Waves and Patterns in Chemical Reactions

Steve Scott

Nonlinear Kinetics Group

School of Chemistry

University of Leeds

steves@chem.leeds.ac.uk

Outline

• Background

• PatternsDIFICIFDO

• Wavesexcitable mediawave block

Feedback

non-elementary processes

intermediate species influence rate of own production and, hence, overall reaction rate.

Waves & PatternsWaves

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

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

Castets et al. Phys Rev. Lett 1990

A. Hunding, 2000

Ouyang and SwinneyChaos 1991

CDIMA reactionTuring Patterns

spots and stripes: depending onExperimentalConditions

“Turing Patterns” in flames

“thermodiffusive instability”

- first observed in Leeds

(Smithells & Ingle 1892)

requires thermal diffusivity < mass diffusivity

DIFICI

• differential-flow induced chemical instability

• still requires selective diffusivity but can be any species

Menzinger and RovinskyPhys. Rev. Lett., 1992,1993

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

p re ssu rereg u la to r

rese rv o ir

io n -ex ch an g eco lu m nlo a d edw ithfe rro in

Experiment

= 2.1 cm

cf = 0.138 cm s1

f = 2.8 s frame1

[BrO3] = 0.8 M

[BrMA] = 0.4 M

[H2SO4] = 0.6 M

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

Experimental results

imaging system: vary “driving pressure”

0.00 0.05 0.10 0.15 0.200.00

0.05

0.10

0.15

0.20

[BrO3

]0 = 0.95 M

[BrO3

]0 = 0.8 M

[BrO3

]0 = 0.6 M

c wa

ve (

cm/s

)

cflow

(cm/s)0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

[BrO3]

0 = 0.95 M

[BrO3]

0 = 0.8 M

[BrO3]

0 = 0.6 M

(c

m)

cf (cm/s)

slope ~ 1

Not possible to determine “critical flow velocity”

BZ reaction

• Involves competition between:

HBrO2 + Br- 2BrMA

and

HBrO2 + BrO3- + 2Mred 2HBrO2 + 2Mox

• Also

BrMA + 2Mox f Br- + 2Mred

Theoretical analysis:

• Dimensionless equations

)(

)()1(

2

2

qu

qufvuu

x

u

x

u

t

u

u = [HBrO2], v = [Mox] : take = 0 and f depend on initial reactant concentrations

vux

v

x

v

t

v

2

2

main results• DIFICI patterns in range of operating

conditions separate from oscillations

f

absoluteinstability

convective instab.

noinstability

no instab.

21

21

]H][BrO[

[MA]

3

cr = 0

cr

cr increasing

5 0

01 0 0 0

x

t

Space-time plot showing position of waves

note: initiation site moves down tube

back to dimensional terms :

predict

cf,cr = 1.3 102 cm s1

Forcf,cr = 2.4 102 cm s1

= 0.42 cm

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

p lu g -flo wrea c to r: f ille d w ithg la ss b e ad s

in flo w 1 in flo w 2

CSTR

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

C S T Rd

1

d1

d1

d1

d1

d1

d2

d2

d2

d4

d3

d4

d5

d3

• Explains:

need for “oscillatory batch” reaction

stationary pattern

wavelength = velocity oscill period

• Doesn’t explain

critical flow velocity

other responses observed

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

]I[

]I][MA[

21

211

k

kr

r k2 2 2 [ ][ ]ClO I

23

2233 ]I[

]I][I][ClO[

k

kr

Dimensionless equations

22

2 41

u

uvu

x

u

x

u

t

u

22

2

u

uvu

x

v

x

v

t

v

u = [I], v = [ClO2]:

uniform steady-state is a solution of these equations,

but is it stable?

J. Bamforth et al., PCCP, 2000, 2, 4013

absolute and convective instability

stationary FDO pattern

Relevance to somatogenesis?

Waves in Excitable Media

• What is an “excitable medium?

• Where do they occur?

Excitability

steady state is stable to small perturbations

system sits at a steady state

Large (suprathreshold) perturbations initiate an excitation event.

System eventually recovers but is refractory for some period

Excitability in Chemical Systems

• BZ reaction:

oscillations

targets

Spirals

broken waves ends evolve into spirals

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

O2 inhibition

Inhibited layer due to presence of O2

(O2 favours reduced state!)

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

Analysis

• Can define a “modified stoichiometric factor”, feff:

where 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

1

ffeff

computations

Can compute wavespeed for different O2 concentrations:

see quenching of wave at high O2

computed wave profiles

O2 profile computedby Zhabotinsky:

J. Phys. Chem., 1993

allows computation of wavespeed with depth

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

Biological systems

• wave propagation widespread:

signalling

sequencing of events

co-ordination of multiple cellular responses

• 1D pulse propagation

nerve signal propagation

Electrical Activity in Heart

Cardiac activity and arrhythmia

Electrical signal and contraction propagate across atria and then into ventricles

3D effects

spirals and fibrillationSimple waves may

break due to local reduced excitability:

ischemia

infarction

scarring

actually 3D structures - scrollscanine heart

L. Glass, Physics Today, August 1996

scrolls in the BZ systemCan 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

2D waves on neuronal tissue

Spreading depression wave in chicken retina

(Brand et al., Int. J. Bifurc. Chaos, 1997)

Universal relationship

dispersion relation: relates speed of wave to period or wavelength

Wave Failure and Wave Block

Industrial problem:

“reaction event” propagating in a non-continuous medium:

sometimes fails

Wave Propagation in Heterogeneous Media

Jianbo Wang

Pyrotechnics - SHS

“thermal diffusion” between reactant particles – heat loss in void spaces

Arvind Varma: Sci. Am. Aug, 2000

Myelinated nerve tissue

propagation by “hopping” from one Node to next

Propgn failure occurs in MS

Ca2+ waves

intra- and inter-cellular waves

airway epithelialcells

(Sneyd et al. FASEB J. 1995)

intra- and intercellular waves

Analysis• Some previous work – mainly directed at

determining “critical” (single) gap width

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”

Modelautocatalytic 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 < 0.071;

for k = 0.04, critical gap size = 5.6 units

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.

• 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”

Example of “rules”

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

Di 14 15 16 17 18 19 20 21

Di+1 20 18 16 16 15 15 15 14

Random distribution of 5-unit gapsabsolute critical

spacing = 14 corresponds to mean spacing for void fractn of 0.26

0.1 void fraction has mean spacing = 45

• 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

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