Waves and Patterns in Chemical Reactions Steve Scott Nonlinear Kinetics Group School of Chemistry University of Leeds

Outline Background Patterns DIFICI FDO Waves excitable media wave block

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

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

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 Swinney Chaos 1991 CDIMA reaction Turing Patterns spots and stripes: depending on Experimental Conditions

“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 Rovinsky Phys. 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

Experiment = 2.1 cm c f = cm s  1 f = 2.8 s frame  1 [BrO 3  ] = 0.8 M [BrMA] = 0.4 M [H 2 SO 4 ] = 0.6 M Rita Toth, Attila Papp (Debrecen), Annette Taylor (Leeds)

Experimental results imaging system: vary “driving pressure” slope ~ 1 Not possible to determine “critical flow velocity”

BZ reaction Involves competition between: HBrO 2 + Br - 2BrMA and HBrO 2 + BrO M red 2HBrO 2 + 2M ox Also BrMA + 2M ox f Br - + 2M red

Theoretical analysis: Dimensionless equations u = [HBrO 2 ], v = [M ox ] : take  = 0  and f depend on initial reactant concentrations

main results DIFICI patterns in range of operating conditions separate from oscillations f absolute instability convective instab. no instability no instab.  cr = 0  cr   cr increasing

Space-time plot showing position of waves note: initiation site moves down tube back to dimensional terms : predict c f,cr = 1.3  10  2 cm s  1 For c f,cr = 2.4  10  2 cm s  1 = 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 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

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 + I 2  IMA + I  + H + (2) ClO 2 + I   ClO 2  + ½ I 2 (3) ClO 2  + 4 I  + 4 H +  Cl  + 2 I 2

Dimensionless equations u = [I  ], v = [ClO 2  ]: 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

O 2 -effects on BZ waves propagate BZ waves in thin films of solution under different atmospheres: main point is that O 2 decreases wave speed and makes propagation harder: this effect is more important in thin layers of solution

O 2 inhibition Inhibited layer due to presence of O 2 (O 2 favours reduced state!)

Mechanistic interpretation Modify “Process C” – clock resetting process: M ox + Org  M red + MA. + H + MA.  g Br  MA. + O 2  (  + 1) MA. rate = k 10 (O 2 )V (cf. branched chain reaction) Presence of O 2 leads to enhanced production of Br  which is inhibitor of BZ autocatalysis

Analysis Can define a “modified stoichiometric factor”, f eff : where  is a ratio of the rate coefficients for MA. branching and production of Br  and increases with O 2. Increasing O 2 increases f and makes system less excitable

computations Can compute wavespeed for different O 2 concentrations: see quenching of wave at high O 2

computed wave profiles O 2 profile computed by 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 fibrillation Simple waves may break due to local reduced excitability: ischemia infarction scarring actually 3D structures - scrolls canine heart L. Glass, Physics Today, August 1996

scrolls in the BZ system Can exploit inhibitory effect of O 2 on BZ system to generate scroll waves wave under air then N 2 wave under O 2 then under N 2 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

Ca 2+ waves intra- and inter- cellular waves airway epithelial cells (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”

Model autocatalytic wave with decay A + 2B 3Brate = ab 2 B Crate = 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 D i, this table indicates the minimum value of the next separation if the wave is to propagate throughout DiDi D i

Random distribution of 5-unit gaps absolute critical spacing = 14 corresponds to mean spacing for void fract n of 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