Pattern Formation and Dynamics in Nonequilibrium Systems

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4 editions of this work

In order to investigate this issue in more detail we visualized the activity pattern of the two frequency bands previously identified. We observed that the slow activity is mainly confined within the very left part of the reaction cloud Fig. Nevertheless, for both frequency ranges comparably high values for the mean phase coherence between the rectangle containing the deposition point and the remaining ones were determined, which were indicative of pronounced phase locking.

This finding is highlighted by Videos S3 and S4 , which show the time evolution of the fast activity and the corresponding mean phase coherence values respectively. In particular Video S3 reveals that the fast activity encounters at almost all instances an ordered spatial oscillation. In fact, the precise oscillation mode changes frequently while the experiment evolves, traversing a sequence of different oscillation patterns.

This sequence seems to be repeated in a kind of cycles. However, Fig. Changes are mainly effective on a small spatial scale of the order of the extension of only one rectangle, while on larger scales the synchronization pattern keeps almost constant see Video S4 of the Supplementary Material. Hence, although the spatial oscillation pattern changes continuously see Video S3 and Fig.

Furthermore, we suspect that there is some coupling between the activity of the fast and slow frequency band. This suspicion is corroborated by the fact that the synchronization profile obtained independently for the two frequency ranges covers the same area, although we could not find any significant indication for this hypothesis in terms of e. Additional matters that deserve special attention, for which we have only disclosed qualitative trends, are moisture, diameter and temperature dependences of the pattern formation.

With regard to the effect of water vapour we confirmed the finding of 18 that under moisture free conditions no Liesegang rings were formed. Concerning tube diameter, we found that as it increases a regular ring spatial distribution 18 gives way to the type of patterns discussed in this paper. This might be related to the advent of a non-negligible convection contribution to the reaction-diffusion process. The temperature issue is fundamental for a proper understanding of the phenomenon under study. Both external condition temperature and internal temperature distribution are important.

Overall, we believe that a finely orchestrated multifactor phenomenon, involving temperature and concentration gradients of diverse origin and their interrelations, are the key elements for the understanding of our observations We are dealing with an open system in which the exothermic nature of the chemical reaction provokes both temperature and concentration gradients. Since the system is principally governed by chemical reaction, diffusive fluxes and convective motions the latter admissible for sufficiently large tube diameters , some aspects could probably be rationalised in terms of a reaction-diffusion-convection system 18 , 19 , The consideration of convection is a new feature in studies of gas-phase Liesegang-type pattern formation.

Further experimentation, such as measurement of the internal temperature distribution, particle tracking, quantitative determination of cloud geometry-flux interrelations, amongst others, is required in order to disentangle in detail the question of how reaction, diffusion and convection originate and generate the observed precipitation pattern.

Summing up, our main results are: the visualization of the propagating pattern forming phenomenon, its identification with a reaction-diffusion-convection process, the discovery of a fast deposition mode previously undetected, the stability along the evolution of the reaction cloud of the synchronization between the activity at the deposition region and all the remaining regions of the reaction cloud.

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As indicated in Fig. The light-sheet was placed just in the middle of the tube circumference. The observed precipitation pattern was not influenced by local heating of the laser beam as confirmed by other experiments without laser illumination that showed similar results. Video S1 see Supplementary Material illustrates the typical dynamics of the reaction cloud. Three experiments were carried out with this configuration. Though the here shown data was derived from one of the experiments, similar results hold for the other two experiments as documented by Fig.

Then the zero-lag correlation was determined between all time pairs of these data vectors:. Here we follow an analytical signal approach 23 , 24 , 25 using the Hilbert Transform. Once the instantaneous phases are estimated a possible phase locking between two signals S i t and S j t can be quantified by. A reliable estimation of the mean phase coherence is only possible for mono-component narrow band signals.

A comprehensive overview about the applicability of techniques based on the estimation of instantaneous frequencies can be found in ref. Therefore, we performed such estimates separately for the slow and fast processes based on the data derived from the light intensity measurements.

Michael C. Cross - Publications

For the slow frequency band a segment of data points was studied. The segment length for the R ij estimation of the fast frequency band was data points or equivalently photographs, which was then shifted with a maximal overlap across the recording. Here we focus on the intensities extracted for each rectangle from the images of Video S1. To this end the images were set in the reference frame of the moving reaction cloud.

Then, for each rectangle the average intensity was measured as a function of time. The ensuing time series were then filtered in two frequency bands, the slow band between 0 and 0. For the filtering procedure a 4 th order Butterworth filter 27 was applied in forward and backward direction in order to minimize possible shifting of the signal phases.

However, filtering causes, in particular for the fast frequency bands that the time evolution of the filtered intensities oscillate around zero. High pass filtering eliminates slow frequency contributions and thereby may produce a possible offset.

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As a result the filtered signals can take positive as well as negative values, which is counterintuitive if intensities are under consideration. Therefore, we added to the filtered intensity values the minimal value of the signals derived from all rectangles. Thereafter we divided each of the signals by the highest absolute maximum of all of them.

Hence the intensity values of all time series vary between zero and one. In order to quantify the time stability of the synchronization pattern within the fast frequency band, we estimated an average difference matrix in the following way:.

Pattern Formation and Solitons

On the dynamics of Liesegang-type pattern formation in a gaseous system. Liesegang, R. Chemische Fernwirkung. The Liesegang phenomenon. Accessed: 22 nd October Lagzi, I. Effect of geometry on the time law of Liesegang patterning. Dee, G. Patterns produced by precipitation at a moving reaction front. Antal, T.

Non-equilibrium dynamics

Derivation of the matalon-packter law for Liesegang patterns. Hantz, P. PhD thesis, University of Geneva, Einarsrud, M. Band formation during gaseous diffusion in aerogels. E 57 , — Ostwald, W. Engelmann, Morse, H. Diffusion and saturation in gelatine. Droz, M. Liesegang patterns: studies on the width law. A new universal law for the Liesegang pattern formation. Lebedeva, M. Bifurcation analysis of Liesegang ring pattern formation. Spotz, E. Liesegang ring formation arising from diffusion of ammonia and hydrogen chlorine gases through air.

Thompson, S. Patterns, oscillations, and microtornadoes: extreme events in vapor-to-particle reaction zones. Role of gravity in the formation of Liesegang patterns. Mormann, F.

Submission history

Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients. Physica D , — Davis, G. Nature 21 , Pojman, J. Connective Effects on Chemical Waves. Mechanism and Stability Criteria. Rossi, F. Segmented waves in a reaction-diffusion-convection system. Chaos 22 , Tang, M. Compilation and evaluation of gas phase diffusion coefficients of reactive trace gases in the atmosphere: volume1. Inorganic compounds.

Spiller, L. Gabor, D. Theory of communication. Picinbono, B. On instantaneous amplitude and phase of signals. IEEE Trans. Signal Process. Rosenblum, M. Phase synchronization of chaotic oscillators. Chavez, M. Towards a proper estimation of phase synchronization from time series. Press, W. Numerical Recipes: The art of scientific computing 2 nd edn , Ch. Download references. The author E. A and F. M did the experiments, G. All authors contributed in the planning of the project and contributed to its development, discussed the results and reviewed the manuscript. This work is licensed under a Creative Commons Attribution 4.

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Pattern Formation and Dynamics in Nonequilibrium Systems

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Pattern Formation and Dynamics in Nonequilibrium Systems Pattern Formation and Dynamics in Nonequilibrium Systems
Pattern Formation and Dynamics in Nonequilibrium Systems Pattern Formation and Dynamics in Nonequilibrium Systems
Pattern Formation and Dynamics in Nonequilibrium Systems Pattern Formation and Dynamics in Nonequilibrium Systems
Pattern Formation and Dynamics in Nonequilibrium Systems Pattern Formation and Dynamics in Nonequilibrium Systems
Pattern Formation and Dynamics in Nonequilibrium Systems Pattern Formation and Dynamics in Nonequilibrium Systems
Pattern Formation and Dynamics in Nonequilibrium Systems Pattern Formation and Dynamics in Nonequilibrium Systems
Pattern Formation and Dynamics in Nonequilibrium Systems Pattern Formation and Dynamics in Nonequilibrium Systems

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