Oscillations, Chaos and Pattern Formation in Chemical...

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This course material is supported by the Higher EducationRestructuring Fund

allocated to ELTE by the Hungarian Government

Oscillations, Chaos and Pattern Formation inChemical Systems

Historical Overview

I 1600s Boyle: oscillatory phosphorus ignition

I 1800s Fechner: oscillatory dissolution of copper intophosphoric acid

I 1885 Poinceare: three-body problem (King Oscar’s Prize)

I 1886 Landolt: iodate clock

I 1896 Liesegang: precipitation patterns

I 1910 Luther: chemical waves; Lotka: oscillations in biologicalsystems

I 1920 Bray: oscillations in the iodate–hydrogen-peroxidesystem; Fisher & Kolmogorov: theory of waves

I 1950-1960 Belousov-Zhabotinsky reaction; Turing: theory ofpattern formation; Prigogine & Nicolis: Brusselator model;Lorenz: chaos theory; Ghosh & Chance: glycolytic oscillations

Historical Overview

I 1970-80 Field–Koros–Noyes mechanism; Winfree: spiralwaves; Epstein, De Kepper, Orban: design of chemicaloscillators; Nobel Prize in Chemistry (1977) Ilya Prigogine:non-equilibrium thermodynamics & theory of dissipativestructures

I 1980-90 Swinney: experimental observation of chaos inchemical systems; Rossler& Feigenbaum: chaos theory; DeKepper & Boissonade: experimental observation of stationarypatterns; Field & Gyorgyi: three variable model of BZ chaos;Kuramato: theory of oscillations, waves and synchronization

Historical Overview

I 1990-2000 Ott & Grebogi & Yorke: theory of controllingchaos; Showalter & Gaspar: controlling chaos in chemicalsystems

I 2000-. . . Nobel Prize in Chemistry 2007 Gerhard Ertl:chemical processes and spatial organization on solid surfaces;Yoshida & De Kepper: chemomechanical oscillations; Hudson& Kiss: synchronization in electrochemical systems; Novak &Tyson: generic cell cycle model (systems biology); Kondo:pigment patterns; systems chemistry

Nonlinear Systems

I the output of a nonlinear system is not directly proportional tothe input

I the whole differs from the sum of its part

I particular effects can not be assigned to particular causalcomponents

I circular causality, feedback loops

I emergence and unpredictability

I a set of simultaneous equations in which the unknowns appearas variables of a polynomial of degree higher than one or inthe argument of a function which is not a polynomial ofdegree one

Nonlinear Systems

Three main classes of nonlinear problems:

I low-dimensional chaos: small differences in initial conditionsyield widely diverging outcomes for such dynamical systems,rendering long-term prediction impossible in general

I solitons: particle like entities that remain organized during thecourse of the dynamics, e.g. tsunami

I reaction-diffusion phenomena: self-organization in dissipativesystems

Nonlinear Chemical Systems

Clock reactions (positive feedback):

(I. Szalai)

3SO2−3 + IO−3 −→ 3SO2−

4 + I−

IO−3 + 5I− +6H+ −→ 3H2O + 3I2 (slow)I2 + H2O + SO2−

3 −→ SO2−4 + 2I− + 2H+ (fast)

Nonlinear Chemical SystemsOscillatory reactions (coupled positive & negative feedback):The Belousov-Zhabotinsky reaction

(I. Szalai)

Nonlinear Chemical SystemsChaotic oscillations in the Belousov-Zhabotinsky reaction:

(reproduced from Gyorgyi et al J. Phys. Chem. 1992, 96, 1228-1233.)

Chemical waves (nonlinear kinetics & transport): Target patternsin the Belousov-Zhabotinsky reaction

(I. Szalai)

Stationary patterns (nonlinear kinetics & differential transport):Bromate-sulfite-ferrocyanide system

(I. Szalai)

Mathematical Background

A dynamical system is a rule for time evolution on a state space. Itcan be described by an initial value problem.An example:a deterministic, autonomous system with continuous time

x = f (x)

x(0) = x0

x ∈ Rn

It has a unique solution if f is continuously differentiable.

Fixed points

Fixed points (stationary states) are an important class of solutionsof differential equations.

f (x∗) = 0

The fixed point x∗ is stable if a solution x(t) based nearby remainsclose to x∗ for all time. If x(t)→ x∗ as t →∞ than x∗ isasymptotically stable.

Fixed points stability in 1D

Linearizing about a fixed point

f (x∗) = 0

∆x = x(t)− x∗

∆x = x

using Taylor’s expansion

f (x∗ + ∆x) = f (x∗) + ∆xf ′(x∗) + O((∆x)2)

if O((∆x)2) terms are negligible, than

∆x = ∆xf ′(x∗)

The perturbation ∆x grows exponentially if f ′(x∗) > 0 and decaysif f ′(x∗) < 0. If f ′(x∗) = 0 than the O((∆x)2) terms are notnegligible.

Bifurcations

The qualitative changes in the dynamics (e.g. fixed points arecreated or destroyed) are called bifurcations, an the parametervalues at which they occur are called bifurcation points.

Saddle-Node bifurcation (the basic mechanism by which fixedpoints are created and destroyed)

x = r + x2

I if r < 0 there are two fixed points (x∗ = ±√r), one stable

(f ′(x∗) = −2√r < 0) and one unstable (f ′(x∗) = 2

√r > 0)

I if r = 0 there is a single fixed point (x∗ = 0)

I if r > 0 there are no fixed point

The bifurcation occurred at r = 0.

Bifurcations

Pitchfork bifurcation

x = rx − x3

I if r < 0 there is a single stable fixed point (x∗ = 0)

I if r = 0 there is a single stable (linearization vanishes) fixedpoint (x∗ = 0)

I if r > 0 there are there are three fixed points: the x∗ = ±√r

are stable, but the x∗ = 0 is unstable. The system is bistable.

The bifurcation occurred at r = 0.

Bifurcations

Imperfect pitchfork bifurcation

x = h + rx − x3

if r > 0 there is hysteresis between the two stable states

(I. Szalai)

Linearizing about a fixed point

x = f (x , y)

y = g(x , y)

fixed points

f (x∗, y∗) = 0

g(x∗, y∗) = 0

perturbations

∆x = x − x∗

∆y = y − y∗

The linearized system:(∆x

(∂f∂x

∂f∂y

∂g∂x

∂g∂y

)(x∗,y∗)

(∆x∆y

)where the matrix

J(x∗,y∗) =

(∂f∂x

∂f∂y

∂g∂x

∂g∂y

)(x∗,y∗)

is called the Jacobian matrix at the fixed point.

The stability of the fixed points is characterized by the eigenvalues(λ1, λ2) of the Jacobian matrix, since the general solution of thelinearized system can be written as:(

∆x∆y

)= c1e

)+ c2e

)The characteristic equation is:

λ2 − τλ+ ∆ = 0

τ = trace(J(x∗,y∗))

∆ = det(J(x∗,y∗))

(source: http://www.scholarpedia.org/article/Equilibrium)

An inherently nonlinear phenomena: limit cycle

A limit cycle is an isolated (the neighboring trajectories are notclosed) closed trajectory.

(I. Szalai)

Andronov-Hopf bifurcation

Supercritical

(I. Szalai)

I The size of the limit cycle grows from zero, and increasesproportional to

√µ− µc

I The period is T = 2πImλ

Bistability and Oscillations

u = −u3 + µ0u − λ− κv (activator)

τ(u − v) (inhibitor)

Cross-shaped diagram

Autocatalysis

Quadratic autocatalysis

A + Bk1−−→ 2B

v1 = k1[A][B]

dt= −k1[A][B]

ξ =[A]0 − [A]

[A]0dξ

dt= k1ξ(1− ξ)

Autocatalysis

Cubic autocatalysis

A + 2Bk2−−→ 3B

v2 = k2[A][B]2

dt= −k1[A][B]2

ξ =[A]0 − [A]

[A]0dξ

dt= k1ξ

2(1− ξ)

Autocatalysis

Example

IO−3 + 5 I− + 6H+ k3−−→ 3 I2 + 3H2O

H3AsO3 + I2 + H2Ok4−−→ H3AsO4 + 2 I− + 2H+

overall stoichiometry

IO−3 + 5 I− + 2H3AsO3 −−→ 3H3AsO4 + 6 I−

d [I−]

dt= (k3 + k4[I−])[I−][IO−3 ][H+]2

Autocatalysis in a CSTRCSTR: continuous flow stirred-tank reactor

(source: commons.wikimedia.org)

A + 2Bk2−−→ 3B

v2 = k2[A][B]2

dt= −k2[A][B]2 + k0([A]0 − [A])

[A]0 + [B]0 = [A] + [B]

dt= −k2[A]([A]0 + [B]0 − [A])2 + k0([A]0 − [A])

Autocatalysis in a CSTR

dt= −k2[A]([A]0 + [B]0 − [A])2 + k0([A]0 − [A])

(I. Szalai)

Oscillations in a CSTR

A + 2Bk2−−→ 3B

v2 = k2[A][B]2

Bk3−−→ C

v3 = k3[B]

dt= −k2[A][B]2 + k0([A]0 − [A])

dt= k2[A][B]2 − k3[B] + k0([B]0 − [B])

Oscillations in a CSTR

A + 2Bk2−−→ 3B

Bk3−−→ C

k2=1000mol−2dm−6s−1, k3=0.2s−1, k0 = 6.67× 10−3 s−1,[A]0=0.1M, [B]0=0.05M

Systematic Design of Chemical Oscillators

Applications of cross-shaped diagram technique

I finding a system that shows bistability (positive feedback)

I find an additional reactant that provides the necessarynegative feedback

I increase the influence of the negative feedback until bistabilitydisappear and oscillation develop

Design of a pH-oscillatorthe bistable subsystem:

SO2−3 + H+ HSO−3

H2O2 + HSO−3 → H+ + SO2−4 + H2O

the negative feedback:

H2O2 + 2[Fe(CN)6]4− + 2H+ → 2H2O + 2[Fe(CN)6]3−

Oscillations in Batch

The Belousov-Zhabotinsky reaction

BrO–3 + Br– + 2 H+ −−→ HBrO2 + HOBr

HBrO–2 + Br– + H+ −−→ 2 HOBr

HOBr + Br– + H+ −−→ Br2 + H2OBr2 + CH2(COOH)2 −−→ BrCH(COOH)2 + H+ + Br–

BrO–3+2 Br–+3 CH2(COOH)2+3 H+ −−→ 3 BrCH(COOH)2+3 H2O

BrO–3 + HBrO2 + H+ −−→ 2 BrO2 · + H2O

BrO2 · + Ce3+ + H+ −−→ HBrO2 + Ce4+

2 HBrO2 −−→ BrO–3 + HOBr + H2O

HOBr + CH2(COOH)2 −−→ BrCH(COOH)2 + H2O

BrO–3 + 4 Ce3+ + CH2(COOH)2 + 5 H+ −−→

BrCH(COOH)2 + 4 Ce4+ + 3 H2O

6 Ce4+ + CH2(COOH)2 + 2 H2O −−→HCOOH + 6 Ce3+ + 2 CO2 + 6 H+

4 Ce4+ + BrCH(COOH)2 + 2 H2O −−→Br– + HCOOH + 4 Ce3+ + 2 CO2 + 5 H+

Oregonator

Y−→X

X + Y−→X−→2X + 2Z

2X−→Z−→fY

(I. Szalai)

Deterministic Chaos

I Chaos is aperiodic long-term behavior in a deterministicsystem that exhibits sensitive dependence on initial conditions

I The trajectories converge to strange attractor, which are oftenfractal sets

I The irregular behavior arises from the nonlinearity

I Nearby trajectories separate exponentially

(source: commons.wikimedia.org)

Chaos in the Belousov-Zhabotinsky Reaction

∆x = δxeλt

D0 ≥ 2 +λ

|λ′|

ExcitabilityI An excitable system has a stable resting stateI The response to a small perturbation is smallI The response to a sufficiently large perturbation (above a

threshold) is qualitatively different: a large amplitudeexcursion in at least one of the state variable before return toth resting state

(I. Szalai)

Reaction-Diffusion Systems

∂t= f(c) + D∆c (boundary condition)

∂x2+

∂y2+

)I Fronts in a bistable system

I Waves in an excitable system

I Stationary patterns in activator-inhibitor systems in presenceof differential diffusion

Fronts

∂t= κu(u − 1) + D

The speed of the propagating front is (Fisher, Kolmogorov andcoworkers):

v ≥ 2√κD

(I. Szalai)

Waves (spirals) in Excitable Systems

v = v∞ −D

rcr =D

v(T ) = v∞ tanh(T

T ∗)

Turing patterns

∂t= f (u, v) + Du

∂t= g(u, v) + Dv

The stability of the homogeneous stationary state can be describedby linearization (the wavenumber of the perturbation is k):

||J−Dk2 − Iω|| = 0

(a bc d

(Du 00 Dv

)and I is the identity matrix.

Turing patternsLet us assume that the homogeneous stationary state is stable anda > 0 (u is an activator) while d < 0 (v is inhibitor of u).The corresponding length-scales are:

lu =√Du/a

lv =√

Dv/(−d)

The condition of the development of stationary patterns is:

l2u− 1

√ad − bc

The intrinsic wavelength of the patterns is:

√ad − bc

λc = 2π/kc

Turing patterns

Turing patterns in ChemistryReversible complexation of the activator (X) by a large molecule

S + X −−⇀↽−− SX

The effective diffusion coefficient:

DeffX = DX/σ

whereσ = 1 + KSB[S]tot

(I. Szalai)

Recommended Textbooks

I A. C. Scott: The Nonlinear UniverseSpringer-Verlag, Berlin Heidelberg, 2007

I S. H. Strogatz: Nonlinear Dynamics and ChaosWestview Press, USA, 1994

I I. R. Epstein, J. Pojman: An Introduction to NonlinearChemical DynamicsOxford University Press, New York, 1988

I P. Gray, S. K. Scott: Chemical Oscillations and InstabilitiesClarendon Press, Oxford, 1990

I Some of the figures and movies have made by Istvan Szalai(Laboratory of Nonlinear Dynamics, Eotvos L. University)

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