CHEMICAL TESSELLATIONS — RESULTS OF BINARY AND TERTIARY REACTIONS BETWEEN METAL IONS AND FERRICYANIDE OR FERROCYANIDE LOADED GELS

2010 ◽  
Vol 20 (07) ◽  
pp. 2241-2252 ◽  
Author(s):  
B. P. J. DE LACY COSTELLO ◽  
I. JAHAN ◽  
P. HAMBIDGE ◽  
K. LOCKING ◽  
D. PATEL ◽  
...  
Keyword(s):  
Metal Ions ◽  
Voronoi Diagram ◽  
Metal Salt ◽  
Voronoi Diagrams ◽  
Reaction Rates ◽  
Cross Reactivity ◽  
Diffusion Reaction ◽  
Reactive Metal ◽  
Simple Chemical ◽  
Pattern Forming

In our recent letter [de Lacy Costello et al., 2009] we described the formation of spontaneous complex tessellations of the plane constructed in simple chemical reactions between drops of metal salts and ferricyanide or ferrocyanide loaded gels. In this paper, we provide more examples of binary tessellations and extend our analysis to tessellations constructed via tertiary mixtures of reactants. We also provide a classification system which describes the tessellation based on the reactivity of the metal salt with the substrate and also the cross-reactivity of the primary products. This results in balanced tessellations where both reactants have equal reactivity or unbalanced tessellations where one reactant has a lower reactivity with the gel. The products can also be partially or fully cross reactive which gives a highly complex tessellation. The tessellations are made up of colored cells (corresponding to different metal ferricyanides or ferrocyanides) separated by bisectors of low precipitate concentration. The tessellations constructed by these reactions constitute generalized Voronoi diagrams. In the case of certain binary or tertiary combinations of reactants where the diffusion/reaction rates differ, then multiplicatively weighted crystal growth Voronoi diagrams are constructed. Where one reactant has limited or no reactivity with the gel (or the products are cross reactive) then the fronts originating from the reactive metal ions cross the fronts originating from the partially reactive metal ions. The fronts can annihilate in the formation of a second Voronoi diagram relating to the relative positions of the reactive drops. Therefore, two or more generalised or weighted Voronoi diagrams can be calculated in parallel by these simple chemical systems. However when these reactions were used to calculate an additively weighted Voronoi diagram (the reaction was initiated at different time intervals) the diagram constructed did not correspond to the theoretical calculation. We use the failure of these reactions to construct an additively weighted Voronoi diagram to prove a mechanism of substrate competition for bisector formation. These tessellations are an important class of pattern forming reactions and are useful in modeling natural pattern forming phenomena in addition to being a great resource for scientific demonstrations.

Calculating Voronoi Diagrams Using Simple Chemical Reactions

2015 ◽  
Vol 25 (01) ◽  
pp. 1540003 ◽  
Author(s):  
Ben De Lacy Costello
Keyword(s):  
Chemical Reactions ◽  
Voronoi Diagrams ◽  
Computation Time ◽  
Cross Reactivity ◽  
Relative Strength ◽  
Specific Chemical ◽  
Natural Systems ◽  
Simple Chemical ◽  
Chemical Inputs ◽  
Pattern Forming

This paper overviews work on the use of simple chemical reactions to calculate Voronoi diagrams and undertake other related geometric calculations. This work highlights that this type of specialised chemical processor is a model example of a parallel processor. For example increasing the complexity of the input data within a given area does not increase the computation time. These processors are also able to calculate two or more Voronoi diagrams in parallel. Due to the specific chemical reactions involved and the relative strength of reaction with the substrate (and cross-reactivity with the products) these processors are also capable of calculating Voronoi diagrams sequentially from distinct chemical inputs. The chemical processors are capable of calculating a range of generalised Voronoi diagrams (either from circular drops of chemical or other geometric shapes made from adsorbent substrates soaked in reagent) , skeletonisation of planar shapes and weighted Voronoi diagrams (e.g., additively weighted Voronoi diagrams, Multiplicitavely weighted Crystal growth Voronoi diagrams). The paper will also discuss some limitations of these processors. These chemical processors constitute a class of pattern forming reactions which have parallels with those observed in natural systems. It is possible that specialised chemical processors of this general type could be useful for synthesising functional structured materials.

scholarly journals Sequential Voronoi Diagram Calculations using Simple Chemical Reactions

2011 ◽  
Vol 3 (3) ◽  
pp. 29-41
Author(s):  
B. P. J. de Lacy Costello ◽  
I. Jahan ◽  
A. Adamatzky
Keyword(s):  
Voronoi Diagram ◽  
Chemical Reactions ◽  
Voronoi Diagrams ◽  
Cross Reactivity ◽  
Potassium Ferricyanide ◽  
Potassium Ferrocyanide ◽  
Metal Salts ◽  
Unconventional Computing ◽  
Simple Chemical ◽  
Calculation Results

In the authors’ recent paper (de Lacy Costello et al., 2010) the authors described the formation of complex tessellations of the plane arising from the various reactions of metal salts with potassium ferricyanide and ferrocyanide loaded gels. In addition to producing colourful tessellations these reactions are naturally computing generalised Voronoi diagrams of the plane. The reactions reported previously were capable of the calculation of three distinct Voronoi diagrams of the plane. As diffusion coupled with a chemical reaction is responsible for the calculation then this is achieved in parallel. Thus an increase in the complexity of the data input does not utilise additional computational resource. Additional benefits of these chemical reactions are that a permanent record of the Voronoi diagram calculation (in the form of precipitate free bisectors) is achieved, so there is no requirement for further processing to extract the calculation results. Previously it was assumed that the permanence of the results was also a potential drawback which limited reusability. This paper presents new data which shows that sequential Voronoi diagram calculations can be performed on the same chemical substrate. This is dependent on the reactivity of the original reagent and the cross reactivity of the secondary reagent with the primary product. The authors present the results from a number of binary combinations of metal salts on both potassium ferricyanide and potassium ferrocyanide substrates. The authors observe three distinct mechanisms whereby secondary sequential Voronoi diagrams can be calculated. In most cases the result was two interpenetrating permanent Voronoi diagrams. This is interesting from the perspective of mapping the capability of unconventional computing substrates. But also in the study of natural pattern formation per se.

ON MULTITASKING IN PARALLEL CHEMICAL PROCESSORS: EXPERIMENTAL FINDINGS

2003 ◽  
Vol 13 (02) ◽  
pp. 521-533 ◽  
Author(s):  
B. P. J. DE LACY COSTELLO ◽  
A. I. ADAMATZKY
Keyword(s):  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
Classical Problem ◽  
Physical Space ◽  
Reaction Diffusion ◽  
Cross Reactivity ◽  
Chemical System ◽  
Concentration Data ◽  
Reaction Diffusion Systems ◽  
Single Chemical

A parallel chemical processor is a thin-layer of a reagent mixture which reacts to changes in its concentration — data configuration — in a predictable way to form a stationary pattern corresponding to the concentration of the reagent — result configuration. A computation in the chemical processor is implemented via the spreading and interaction of diffusive or phase waves. We design chemical processors that solve a classical problem of computational geometry — computation of a Voronoi diagram. Namely, we study the possibility of designing a multitasking chemical processor that independently and simultaneously computes Voronoi diagrams of two different data planar sets. We define a two-tasking chemical processor as two distinct reactant–substrate couples within a reaction–diffusion processor that solve separate tasks but share the same physical space. A micro-volume of the physical space is an elementary processor of a massively parallel chemical processor, therefore two reaction–diffusion systems occupying the same space are considered to be a single chemical processor. We found that when a single reactant is on a gel layer containing either one or two substrates the same single Voronoi diagram corresponding to the original location of the reactant drops is constructed. However, when two reactants are on a gel containing two substrates and where there is extremely limited cross reactivity between the separate reactant-substrate couples then two Voronoi diagrams of the data planar points (two sets of drops of separate reactants) are constructed; the third "complementary" pattern is also constructed. The first Voronoi diagram constructed is identical at least in position to the one constructed where one reactant was with one substrate (with the same original configuration of reactant drops). After the formation of the first diagram is completed the diffusion fronts corresponding to unlike reactants cross and are only annihilated where they meet another reactant front composed of the same reactant. The result is the computation of two additional Voronoi diagrams pertaining to the spatial positions of the two sets of reactant drops. The outcomes of this experiment albeit in a simple chemical system are significant because the system constitutes the first class of a synthetic chemical parallel processor capable of at least two computations at the same time.

scholarly journals Time-Space Characterization of Wellbore-Cement Alteration by CO2-Rich Brine

Geosciences ◽  
2018 ◽  
Vol 8 (12) ◽  
pp. 490
Author(s):  
Maria Garcia-Rios ◽  
Philippe Gouze
Keyword(s):  
Numerical Models ◽  
Reaction Rates ◽  
Diffusion Reaction ◽  
Cement Matrix ◽  
Relative Distribution ◽  
Reaction Fronts ◽  
Time Space ◽  
Mass Transfers ◽  
Dissolution And Precipitation ◽  
Diffusion Properties

The risk of CO2 leakage from damaged wellbore is identified as a critical issue for the feasibility and environmental acceptance of CO2 underground storage. For instance, Portland cement can be altered if flow of CO2-rich water occurs in hydraulic discontinuities such as cement-tubing or cement-caprock interfaces. In this case, the raw cement matrix is altered by diffusion of the solutes. This fact leads to the formation of distinctive alteration fronts indicating the dissolution of portlandite, the formation of a carbonate-rich layer and the decalcification of the calcium silicate hydrate, controlled by the interplay between the reaction kinetics, the diffusion-controlled renewing of the reactants and products, and the changes in the diffusion properties caused by the changes in porosity induced by the dissolution and precipitation mechanisms. In principle, these mass transfers can be easily simulated using diffusion-reaction numerical models. However, the large uncertainties of the parameters characterizing the reaction rates (mainly the kinetic and thermodynamic coefficients and the evolving reactive surface area) and of the porosity-dependent diffusion properties prevent making reliable predictions required for risk assessment. In this paper, we present the results of a set of experiments consisting in the alteration of a holed disk of class-G cement in contact with a CO2-rich brine at reservoir conditions (P = 12 MPa and T = 60 °C) for various durations. This new experimental protocol allows producing time-resolved data for both the spatially distributed mass transfers inside the cement body and the total mass transfers inferred from the boundary conditions mass balance. The experimental results are used to study the effect of the fluid salinity and the pCO2 on the overall reaction efficiency. Experiments at high salinity triggers more portlandite dissolution, thinner carbonate layers, and larger alteration areas than those at low salinity. These features are accompanied with different spatial distribution of the alteration layers resulting from a complex interplay between salinity-controlled dissolution and precipitation mechanisms. Conversely, the effect of the pCO2 is more intuitive: Increasing pCO2 results in increasing the overall alteration rate without modifying the relative distribution of the reaction fronts.

scholarly journals Correlation between Functional Group and Formation of Nanoparticles in PEBAX/Ag Salt/Al Salt Complexes for Olefin Separation

Polymers ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 667
Author(s):  
So Young Kim ◽  
Younghyun Cho ◽  
Sang Wook Kang
Keyword(s):  
Fourier Transform ◽  
Metal Ions ◽  
Polymer Matrix ◽  
Functional Group ◽  
Metal Salt ◽  
Amide Group ◽  
Separation Performance ◽  
Ether Group ◽  
Long Time ◽  
Ft Ir

poly ether-block-amide (PEBAX)-2533/metal salt/Al salt membranes were prepared for mixed olefin/paraffin separation. PEBAX-2533 with 80% ether group and 20% amide group was suggested as the polymer matrix for comparison of separation performance according to the functional group ratio in copolymer PEBAX. In addition, Al salts were used to stabilize metal ions for a long time as additives. High permeance was expected with the proportion of high ether groups, since these functional groups provided relatively permeable regions. As a result, the PEBAX-2533 composite membrane showed a selectivity of 5 (propylene/propane) with 10 GPU. However, the permeance of membrane was not unexpectedly improved and the selectivity was reduced. The result was analyzed by using SEM, RAMAN and thermogravimetric analysis (TGA), including Fourier transform infrared (FTIR). The reduction in separation performance was determined by using FT-IR. Based on these results, in order to stabilize the metal ions interacting with the polymer through Al(NO3)3, it was concluded that a specific ratio of the amide group was needed in PEBAX as a polymer matrix.

Discrete Construction of Compoundly Weighted Voronoi Diagram

2013 ◽  
Vol 467 ◽  
pp. 545-548
Author(s):  
Hui Wang
Keyword(s):  
Production Process ◽  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
High Potential ◽  
Graphic Display ◽  
Potential Value ◽  
Discrete Algorithms ◽  
Traditional Algorithm ◽  
Definition Of

Compoundly weighted Voronoi diagram is difficult to construct because the bisector is fairly complex. In traditional algorithm, production process is always extremely complex and it is more difficult to graphic display because of the complex definition of mathematic formula. In this paper, discrete algorithms are used to construct compoundly weighted Voronoi diagrams. The algorithm can get over all kinds of shortcomings that we have just mentioned. So it is more useful and effective than the traditional algorithm. The results show that the algorithm is both simple and useful, and it is of high potential value in practice.

VORONOI DIAGRAMS FOR A TRANSPORTATION NETWORK ON THE EUCLIDEAN PLANE

2006 ◽  
Vol 16 (02n03) ◽  
pp. 117-144 ◽  
Author(s):  
SANG WON BAE ◽  
KYUNG-YONG CHWA
Keyword(s):  
Voronoi Diagram ◽  
Euclidean Plane ◽  
Transportation Network ◽  
Voronoi Diagrams ◽  
The Other ◽  
Constant Number ◽  
Time Path ◽  
The Given

This paper investigates geometric and algorithmic properties of the Voronoi diagram for a transportation network on the Euclidean plane. In the presence of a transportation network, the distance is measured as the length of the shortest (time) path. In doing so, we introduce a needle, a generalized Voronoi site. We present an O(nm2+ m3+ nm log n) algorithm to compute the Voronoi diagram for a transportation network on the Euclidean plane, where n is the number of given sites and m is the complexity of the given transportation network. Moreover, in the case that the roads in a transportation network have only a constant number of directions and speeds, we propose two algorithms; one needs O(nm + m2+ n log n) time with O(m(n + m)) space and the other O(nm log n + m2log m) time with O(n + m) space.

THE METHOD OF INTER-PARTICLE DISTRIBUTION FUNCTIONS FOR DIFFUSION-REACTION SYSTEMS IN ONE DIMENSION

1995 ◽  
Vol 09 (15) ◽  
pp. 895-919 ◽  
Author(s):  
DANIEL BEN-AVRAHAM
Keyword(s):  
Contact Process ◽  
Reaction Rates ◽  
Distribution Functions ◽  
Diffusion Reaction ◽  
Nonequilibrium Dynamics ◽  
One Dimension ◽  
Many Body ◽  
Dynamical Phase ◽  
Diffusion Limited ◽  
Diffusion Controlled

Diffusions limited reactions in confined geometries exhibit all aspects of nonequilibrium dynamics, such as anomalous kinetics, self-organization and dynamical phase transitions, and have therefore been the subject of extensive research in recent years. In this paper we review the method of interparticle distribution functions (IPDF) which was originally introduced for deriving the exact kinetics of the diffusion-limited coalescence process, A+A→A, in one dimension. We explain the IPDF method and review variants of the coalescence model which can be solved exactly through this technique. We then consider strategies for approximations based on the IPDF method and review applications to coalescence with finite reaction rates (away from the strictly diffusion-controlled regime), many-body reactions (nA→mA), and the contact process. We conclude with a discussion of open, interesting problems and possible ways to their solution.

scholarly journals On Voronoi Diagrams on the Information-Geometric Cauchy Manifolds

Entropy ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. 713 ◽  
Author(s):  
Frank Nielsen
Keyword(s):  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
Information Geometry ◽  
Square Root ◽  
Chi Square ◽  
Scale Parameters ◽  
Leibler Divergence ◽  
Finite Set ◽  
Metric Distance ◽  
Rao Distance

We study the Voronoi diagrams of a finite set of Cauchy distributions and their dual complexes from the viewpoint of information geometry by considering the Fisher-Rao distance, the Kullback-Leibler divergence, the chi square divergence, and a flat divergence derived from Tsallis entropy related to the conformal flattening of the Fisher-Rao geometry. We prove that the Voronoi diagrams of the Fisher-Rao distance, the chi square divergence, and the Kullback-Leibler divergences all coincide with a hyperbolic Voronoi diagram on the corresponding Cauchy location-scale parameters, and that the dual Cauchy hyperbolic Delaunay complexes are Fisher orthogonal to the Cauchy hyperbolic Voronoi diagrams. The dual Voronoi diagrams with respect to the dual flat divergences amount to dual Bregman Voronoi diagrams, and their dual complexes are regular triangulations. The primal Bregman Voronoi diagram is the Euclidean Voronoi diagram and the dual Bregman Voronoi diagram coincides with the Cauchy hyperbolic Voronoi diagram. In addition, we prove that the square root of the Kullback-Leibler divergence between Cauchy distributions yields a metric distance which is Hilbertian for the Cauchy scale families.

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