Morphology of Experimental and Simulated Turing Patterns

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2.2 Lengyel-Epstein (LE) model for the CIMA reactionThe Lengyel-Epstein (LE) model is based on the stoichiometric reaction equationsthat are believed to describe the CIMA reaction. The region of Turing pattern formationin the parameter space of the LE model can be derived using linear stabilityanalysis.2.2.1 Derivation of the reaction-diffusion equationsA theoretical model for the chemical mechanism behind the CIMA reaction wasproposed by Lengyel and Epstein in 1992 [27]. A brief overview of the model thatcovers all the important aspects is given in Ref. [54], while a detailed description isgiven in Ref. [28].Lengyel and Epstein identified the key reactants in the system and the crucial roleof the starch indicator as a complexing agent, i.e. it weakly binds iodide and iodineto slow down the effective diffusion rates, see also Ref. [1]. An underlying reaction inthe CIMA experiment, the chlorine-dioxide-iodine-malonic acid (CDIMA) reactionis responsible for the pattern formation.In the CIMA reaction chlordioxide and iodine are produced as intermediates withnear constant concentration and the process is again described by the CDIMA reaction.Consequently both reactions are mathematically equivalent.The reaction is described by three stoichiometric 1 equations of the five independentchemical ingredients MA, I 2, ClO 2, ClO − 2 and I− as shown in Ref. [30] and Ref. [29].The first is the reaction of malonic acid (MA) and iodine (I 2 ):MA + I 2 −→ IMA + I − + H + with rate r 1 = k 1a[MA][I 2 ]. (2.1)k 1b + [I 2 ]The second is a reaction between chlorine dioxide (CLO 2 ) and iodide (I − ):ClO 2 + I − −→ ClO − 2 + 1 2 I 2 with rate r 2 = k 2 [ClO 2 ][I − ], (2.2)and the third component reaction between chlorite (ClO − 2 ) and iodide (I− ) isClO − 2 + 4I− + 4 H + −→ 2 I 2 + Cl − + 2 H 2 Owith rate r 3 = k 3a [ClO − 2 ][I− ][H + ] + k 3b[ClO − 2 ][I 2 ][I− ]α + [I − ] 2 . (2.3)Cl − and IMA are inert products of the process, while the concentration [H + ] istreated as constant. Concentrations of any reactant are written with square brackets[]. A reversible complexation 2 of iodine and iodide results in a slower effectivediffusion rate for the activator iodide, as the starch-triiodide complex is immobile1 An introduction to stoichiometry is found in Ref. [65]2 The formation of chemical complexes, see Ref. [65].20
in the gel. This mechanism results in a fourth reaction equation that has to beconsidered:S + I 2 + I − −−⇀ ↽−− SI − 3 with rate r 4 = k 4 [S][I 2 ][I − ] − k 4− [SI − 3 ]. (2.4)where [S] represents the concentration of complexing sites, e.g. the concentrationof starch in the gel. The effective rate laws for r 1 and r 3 account for complicatedautocatalytic intermediate steps in the full reaction mechanism [9,22,29].Eq. (2.1-2.3) together with Eq. (2.4) yields a seven component reaction-diffusionsystem. However for a wide range of experimental conditions [ClO − 2 ] and [I− ] changerapidly by several orders of magnitude, while [ClO 2 ], [I 2 ] and [MA] vary much moreslowly [26]. Treating the slowly varying concentrations as constant and assumingthat the concentration of starch is large and uniformly distributed in the gel, themodel can be reduced to a two variable reaction-diffusion system, referred to as theLengyel-Epstein (LE) model:α −→ U r M1 = k 1 ′ k 1 ′ = k 1a[MA][I 2 ], (2.5)k 1b + [I 2 ]U −→ V r M2 = k ′ 2 [U] k′ 2 =k 2[ClO 2 ], (2.6)4U + V −→ Ω r M3 = k 3′ [U][V]α + [U] 2 k 3 ′ =k 3b[I 2 ], (2.7)S + U −−⇀ ↽−− SU r M4 = k ′ 4 [U] − k 4−[SU] k ′ 4 =k 4[S][I 2 ], (2.8)with U = I − and V = ClO − 2 . Symbolically α denotes the constant reactants and Ωthe inert products. SU is the chemical complex formed by starch and iodide. Theconcentrations [S] and [A] are constant and [SU] is assumed to be a linear functionof [U] for large concentrations of S. In the rate equation for r M3 the k 3a term fromEq. (2.3) has been neglected [29]. The resulting reaction diffusion equations are∂[U]= k 1 ′ − k ′∂t2[U] − k 4[SU] ′ − 4k 3′ [U][V]α + [U] 2 + D U∇ 2 [U], (2.9)∂[V]= k ′∂t2[U] − k 3′ [U][V]α + [U] 2 + D V ∇ 2 [V], (2.10)∂[SU]= k 4 ′ ∂t4−[SU]. (2.11)Adding Eq. (2.9) and Eq. (2.11) and using that for large concentrations of thecomplexing agent S,[SU] = k′ 4k 4−[U] = K ′ [U] with K ′ = k 4k 4−[S][I 2],i.e. [SU] is proportional to [U], one obtains(1 + K′ ) ∂[U]∂t= k 1 ′ − k′ 2 [U] − [U][V]4k′ 3α + [U] 2 + D U∇ 2 [U]. (2.12)21
Page 5: AbstractTuring patterns formed in t
Page 10: 5.5 Minkowski-maps of patterns in t
Page 13 and 14: of the experimental quasi-2D patter
Page 15: (a) “Bacteria” in a reactiondif
Page 18 and 19: Figure 2.1: First evidence for Turi
Page 22 and 23: After rescaling the differential eq
Page 24 and 25: A full derivation is given in Ref.
Page 27 and 28: Chapter 3Numerical solution of 2Dre
Page 29 and 30: The advantage of the FTCS method is
Page 31 and 32: andA u · u n+1 = b n u (u,v) (3.14
Page 33 and 34: withχ u = 2γ u sin 2 k x∆2+ 2γ
Page 35 and 36: 3.4 Convergence to stationary patte
Page 37 and 38: 3.5 Discretization of stochastic pa
Page 39 and 40: Chapter 4Patterns in the Lengyel-Ep
Page 41 and 42: (a) Inverted hexagonalstate: a = 9.
Page 43 and 44: Figure 4.4: Overview of pattern for
Page 45 and 46: 1.61.41.21b0.80.60.40.205 10 15 20
Page 47 and 48: 1.61.41.21b0.80.60.40.205 10 15 20
Page 49: Figure 4.10: The concentration prof
Page 52 and 53: (a) ρ = 0.22, V = 0.77, S =0.15,
Page 54 and 55: Figure 5.2: Illustration for the ma
Page 56 and 57: perimeter S(ρ) and the Euler chara
Page 58 and 59: (a) Experimental hexagonalpattern.
Page 60 and 61: 5.3 Minkowski functionals of simula
Page 62 and 63: (a) Numerical hexagonalpattern, b =
Page 64 and 65: (a) Experimental hexagonalpattern(b
Page 66 and 67: 5.4 Local Minkowski functionals of
Page 68 and 69: P V3210-1-2-1 -0.5 0 0.5 1ρP S2.42
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(a) Experimental hexagonalpattern(b
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5.5 Minkowski-maps of patterns in t
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V(ρ)10.80.60.40.200 0.2 0.4 0.6 0.
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(a) Experimental hexagonalpattern.
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5.7 Minkowski functionals of patter
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Chapter 6Influence of additive nois
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∆u/N 20.2∆u/N 20.1∆u/N 20.10.
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(a) Experimental lamellae.(b) Numer
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Chapter 7Statistical ensemble of su
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HexagonalLamellar0.10.20.51.0amplit
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Experimental patternSuperposed patt
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In this model no dynamics are expli
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Experimental patternSuperposed patt
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7.3 Interacting pattern gasIn this
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7.3.3 Numerical resultsIn this sect
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Chapter 8Summary and outlookA quant
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•••••••••••
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(a)(b)(c)(d)(e)(f)Figure 9.3: Patte
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Appendix ADetails on experimental d
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Bibliography[1] K. Agladze, E. Dulo
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[28] I. Lengyel and J. A. Pojman. A
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[54] B. Rudovics, E. Barillot, P. W
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ErklärungHiermit bestätige ich, d
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Morphology of Experimental and Simulated Turing Patterns

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