Waves and Patterns in Chemical Reactions

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Waves and Patterns in Chemical Reactions

• Steve Scott
• Nonlinear Kinetics Group
• School of Chemistry
• University of Leeds
• steves_at_chem.leeds.ac.uk

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Outline

• Background
• Patterns DIFICI FDO
• Waves excitable media
• wave block

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Feedback

• non-elementary processes
• intermediate species influence rate of own
  production and, hence, overall reaction rate.

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Waves Patterns
Waves uniform steady state localised disturbance leads to propagating front repeated initiation leads to successive waves precise structure depends on location of initiation sites Patterns uniform state is unstable spatial structure develops spontaneously (maybe through waves) pattern robust to disturbance wavelength determined by kinetics/diffusion
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Turing Patterns

• Turing proposal for morphogenesis (1952)
• selective diffusion in reactions with feedback
• requires diffusivity of feedback species to be
  reduced compared to other reactants
• recently observed in experiments
• not clear that this underlies embryo development

A. Hunding, 2000
Castets et al. Phys Rev. Lett 1990
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Ouyang and Swinney Chaos 1991 CDIMA
reaction Turing Patterns spots and stripes
depending on Experimental Conditions
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Turing Patterns in flames

• thermodiffusive instability
• first observed in Leeds
• (Smithells Ingle 1892)
• requires thermal diffusivity lt mass diffusivity

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DIFICI

• differential-flow induced chemical instability
• still requires selective diffusivity but can be
  any species

Menzinger and Rovinsky Phys. Rev. Lett., 1992,1993
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BZ reaction DIFICI

• immobilise ferroin on ion-exchange resin
• flow remaining reactants down tube
• above a critical flow velocity, distinct
  stripes of oxidation (blue) appear and travel
  through tube

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Experiment

• 2.1 cm
• cf 0.138 cm s-1
• f 2.8 s frame-1
• BrO3- 0.8 M
• BrMA 0.4 M
• H2SO4 0.6 M

Rita Toth, Attila Papp (Debrecen), Annette Taylor
(Leeds)
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Experimental results

• imaging system vary driving pressure

slope 1
Not possible to determine critical flow velocity
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BZ reaction

• Involves competition between
• HBrO2 Br- 2BrMA
• and
• HBrO2 BrO3- 2Mred 2HBrO2 2Mox
• Also
• BrMA 2Mox f Br- 2Mred

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Theoretical analysis

• Dimensionless equations

u HBrO2, v Mox take d 0 e and f
depend on initial reactant concentrations
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main results

• DIFICI patterns in range of operating conditions
  separate from oscillations

fcr
convective instab.
no instab.
fcr increasing
fcr 0
no instability
absolute instability
f
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Space-time plot showing position of waves
back to dimensional terms predict cf,cr 1.3
10-2 cm s-1 Forcf,cr 2.4 10-2 cm s-1 l
0.42 cm
note initiation site moves down tube
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Flow Distributed Oscillations

• patterns without differential diffusion or flow
• Very simple reactor configuration plug-flow
  tubular reactor fed from CSTR
• reaction run under conditions so it is
  oscillatory in batch, but steady-state in CSTR

CSTR
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Simple explanation

• CSTR ensures each droplet leaves with same
  phase
• Oscillations occur in each droplet at same time
  after leaving CSTR and, hence, at same place in
  PFR

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• Explains
• need for oscillatory batch reaction
• stationary pattern
• wavelength velocity oscill period
• Doesnt explain
• critical flow velocity
• other responses observed

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CDIMA reaction

• chlorine dioxide iodine - malonic acid
  reaction
• Lengyel-Epstein model
• (1) MA I2 ? IMA I? H
• (2) ClO2 I? ? ClO2? ½ I2
• (3) ClO2? 4 I? 4 H ? Cl? 2 I2

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Dimensionless equations
u I-, v ClO2- uniform steady-state is a
solution of these equations, but is it stable?
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J. Bamforth et al., PCCP, 2000, 2, 4013
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absolute and convective instability
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stationary FDO pattern
Relevance to somatogenesis?
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Waves in Excitable Media

• What is an excitable medium?
• Where do they occur?

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Excitability
system sits at a steady state

• steady state is stable to small perturbations

Large (suprathreshold) perturbations initiate an
excitation event. System eventually recovers but
is refractory for some period
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Excitability in Chemical Systems

• BZ reaction
• oscillations
• targets

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Spirals

• broken waves ends evolve into spirals

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O2-effects on BZ waves

• propagate BZ waves in thin films of solution
  under different atmospheres

main point is that O2 decreases wave speed and
makes propagation harder this effect is more
important in thin layers of solution
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O2 inhibition

• Inhibited layer due to presence of O2

(O2 favours reduced state!)
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Mechanistic interpretation

• Modify Process C clock resetting process
• Mox Org ? Mred MA. H
• MA. ? g Br?
• MA. O2 ? (? 1) MA. rate
  k10(O2)V
• (cf. branched chain reaction)
• Presence of O2 leads to enhanced production of
  Br- which is inhibitor of BZ autocatalysis

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Analysis

• Can define a modified stoichiometric factor,
  feff
• where a is a ratio of the rate coefficients for
  MA. branching and production of Br- and increases
  with O2.
• Increasing O2 increases f and makes system less
  excitable

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computations

• Can compute wavespeed for different O2
  concentrations
• see quenching of wave at high O2

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computed wave profiles
allows computation of wavespeed with depth
O2 profile computed by Zhabotinsky J. Phys.
Chem., 1993
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targets and spirals in flames

• target and spiral structures observed on a
  propagating flame sheet Pearlman, Faraday Trans
  1997 Scott et al. Faraday Trans. 1997

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Biological systems

• wave propagation widespread
• signalling
• sequencing of events
• co-ordination of multiple cellular responses

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nerve signal propagation

• 1D pulse propagation

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Electrical Activity in Heart
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Cardiac activity and arrhythmia

• Electrical signal and contraction propagate
  across atria and then into ventricles
• 3D effects

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spirals and fibrillation

• Simple waves may break due to local reduced
  excitability
• ischemia
• infarction
• scarring

L. Glass, Physics Today, August 1996
actually 3D structures - scrolls
canine heart
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scrolls in the BZ system

• Can exploit inhibitory effect of O2 on BZ system
  to generate scroll waves

wave under air then N2
wave under O2 then under N2
A.F. Taylor et al. PCCP, 1999
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(No Transcript)
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2D waves on neuronal tissue

• Spreading depression wave in chicken retina
• (Brand et al., Int. J. Bifurc. Chaos, 1997)

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Universal relationship

• dispersion relation relates speed of wave to
  period or wavelength

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Wave Failure and Wave Block

• Industrial problem
• reaction event propagating in a non-continuous
  medium
• sometimes fails

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Wave Propagation in Heterogeneous Media
Jianbo Wang
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Pyrotechnics - SHS
Arvind Varma Sci. Am.
Aug, 2000

• thermal diffusion between reactant particles
  heat loss in void spaces

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Myelinated nerve tissue

• propagation by hopping from one Node to next
• Propgn failure occurs in MS

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Ca2 waves
intra- and inter- cellular waves airway
epithelial cells (Sneyd et al. FASEB J. 1995)
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intra- and intercellular waves
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Analysis

• Some previous work mainly directed at
  determining critical (single) gap width

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• We have been interested in a slightly different
  question
• have many gaps randomly distributed, all less
  than critical width

seek to determine critical spacing and
expected propagation success rate
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Model

• autocatalytic wave with decay
• A 2B 3B rate ab2
• B C rate kb
• Assume reactant A is non-uniformly distributed
  where A 0 have gaps
• Only B diffuses decay step occurs even in gaps
• need k lt 0.071
• for k 0.04, critical gap size 5.6 units

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multiple gaps and spacing

• all gaps 5.0
• spacing D varies
• failure occurs if spacing not sufficient to allow
  full recovery of wave between gaps.

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• Have developed a set of rules which allow us to
  judge whether a wave is likely to propagate
  throughout whole of domain on the basis of
  sequence of gap spacings.
• Generate 1000 (say) random gap spacings to
  satisfy some overall void fraction
• Inspect each set to determine whether it passes
  or fails the rules.
• Calculate fraction of passes

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Example of rules
Di 14 15 16 17 18 19 20 21
Di1 20 18 16 16 15 15 15 14

• For a given separation Di, this table indicates
  the minimum value of the next separation if the
  wave is to propagate throughout

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Random distribution of 5-unit gaps

• absolute critical spacing 14 corresponds to
  mean spacing for void fractn of 0.26
• 0.1 void fraction has mean spacing 45

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• Can choose different gap distributions same
  rules, so just need to generate distribution
  sets.
• Could consider random gap widths need to
  develop new rules
• Extend to bistable wave or excitable wave
  dynamics for biological systems

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Acknowledgements

• Matt Davies, Jonnie Bamforth, Jianbo Yang , Alice
  Lazarovici, Phil Trevelyan, Annette Taylor, Barry
  Johnson Leeds
• Rita Toth, Vilmos Gaspar Debrecen
• John Merkin, Serafim Kalliadasis
• British Council Hungarian Academy
• ESF Scientific Programme REACTOR
• EPSRC