1 Hydrodynamic instabilities of autocatalytic reaction fronts: A. De Wit Unité de Chimie Physique non Linéaire Université Libre de Bruxelles, Belgium

2 Scientific questions Can chemical reactions be at the origin of hydrodynamic instabilities (and not merely be passively advected by the flow) ? What are the properties of the new patterns that can then arise ? Influence on transport and yield of reaction ?

3 Outline Convective deformation of chemical frontsConvective deformation of chemical fronts Experiments in Hele-Shaw cellsExperiments in Hele-Shaw cells Model of hydrodynamic instabilities of frontsModel of hydrodynamic instabilities of fronts Rayleigh-Bénard, Rayleigh-Taylor and double- diffusive instabilitiesRayleigh-Bénard, Rayleigh-Taylor and double- diffusive instabilities Reactive vs non-reactive systemReactive vs non-reactive system I. Vertical set-ups II. Horizontal set-ups Marangoni vs buoyancy-driven flows Marangoni vs buoyancy-driven flows

4 Buoyancy-driven instability of a chemical front in a vertical set-up Hydrodynamic Rayleigh- Taylor instability of autocatalytic IAA fronts ascending in the gravity field in a capillary tube because reactant A is heavier than the product B Light B Heavy A: stable descending front Heavy A Light B: unstable ascending front (Courtesy D. Salin) Bazsa and Epstein, 1985; Nagypal, Bazsa and Epstein, 1986 Pojman, Epstein, McManus and Showalter, 1991

5 Model system: fingering of chemical fronts in Hele-Shaw cells Fresh reactants Products Density of reactants different than density of products: ∆  c =  prod -  react ≠0 Buoyantly unstable fronts Carey, Morris and Kolodner, PRE (1996) Böckmann and Müller, PRL (2000) Horvath et al. (2002)

Density of products (top) larger than density of reactants (bottom)  c  c >0 Buoyantly unstable DESCENDING fronts Horvàth et al., JCP (2002) Chlorite-tetrathionate (CT) reaction Böckmann and Müller, PRL (2000) Iodate-arsenous acid (IAA) reaction  c Reactants heavier than products  c <0 : Buoyantly unstable ASCENDING fronts Isothermal system

7 J.A. Pojman and I.R. Epstein, “Convective effects on chemical waves: 1. Mechanisms and stability criteria”, J. Phys. Chem., 94, 4966 (1990)  =  o [1-   (T-T o )+  c  i (C i -C io )] Across a chemical front:  products  reactants  c    c  c  c 0: products are heavier than reactants (CT)  T  T <0: exothermic reaction, products are hotter than reactants

Antagonist solutal and thermal effects Cooperative solutal and thermal effects

9 Questions Which kind of hydrodynamic instabilities can the competition Which kind of hydrodynamic instabilities can the competition or cooperative effects between solutal and thermal density effects generate across a chemical front ? Are there new instabilities possible with regard to the Are there new instabilities possible with regard to the non reactive case ?

10 Theoretical model with Le=D T /D Rayleigh numbers Lewis number -  C,T  g

11 Linear stability analysis of pure hydrodynamic instabilities F(C) = 0 Base state :r concentration and temperature Base state : linear concentration and temperature gradients with a corresponding density gradient

12 Hydrodynamic Rayleigh-Bénard instability Fluid heated from below HOT COLD R T >0

13 Hydrodynamic Rayleigh-Taylor instability Fernandez et al., J. Fluid Mech. (2002) Heavy fluid on top of a light one R c >0

Double diffusive instability Salt fingers: Hot salty water lies over cold fresh water of a higher density. The stratification is kept gravitationally stable. The key to the instability is the fact that heat diffuses much more rapidly than salt (hence the term double-diffusion). A downward moving finger of warm saline water cools off via quick diffusion of heat, and therefore becomes more dense. This provides the downward buoyancy force that drives the finger. Similarly, an upward-moving finger gains heat from the surrounding fingers, becomes lighter, and rises. Salt fingers: Instability even if light solution on top of a heavy solution (statistically stable density gradient) ! R T 0

Pure double diffusion (without chemistry): Le=20 Thermal Rayleigh-Bénard Solutal Rayleigh-Taylor UNSTABLE STABLE OSCILLATING Heated from below Cooled from below Light at the bottom Heavy at bottom Salt fingers Turner

16 Chemical fronts F( C ) = - C (C-1) (C+d)

Base state for the linear stability analysis: reaction-diffusion fronts for both concentration and temperature, connecting the reactants where (C,T)=(0,0) to the products for which (C,T)=(1,1) and traveling at a speed v v Reactants at room temperature Hot products g T profile is function of Le F( C ) = - C (C-1) (C+d)

Convection with chemistry : Le = 1 1: light but cold on top of heavy but hot: unstable if sufficiently exothermic Thermal plumes 2: heavy and cold on top of light and hot: always unstable 3: heavy but hot on top of light but cold: stable descending fronts if sufficiently exothermic Ascending Descending Heavier reactants Lighter reactants Exothermic reaction IAA IAA CT CT

Stable UNSTABLE UNSTABLE

20 Instability due to thermal diffusion and chemistry Descending exothermic front Light and hot products Heavy and cold reactants F(C 1 ) < F(C 2 ) C 1,T 1 The little protrusion reaches rapid thermal equilibrium but still reacts at a rate F(C 1 ) smaller than the rate F(C 2 ) of its surroundings. It gains thus less heat (the reaction being exothermic) and it thus continues to sink. C 2,T 2 C1,T2C1,T2C1,T2C1,T2 T1>T2T1>T2T1>T2T1>T2 Le>1 g

21 Properties of this instability Because the region with F’(c) >0 is followed by a region with F’(c) <0, the local instability is constrained by the region of local stability. This unusual instability magnifies with a larger negative R T and larger Le since  (x) = -R T T -R c C Light and hot Heavy and cold g

Stable Rayleigh-Taylor (heavy over light) New instability of light over heavy

Antagonist solutal and thermal effects: double diffusive instabilities Cooperative solutal and thermal effects: candidates for the new instability for descending fronts

Conclusions and perspectives Classification of the various hydrodynamic instabilities of exothermic reaction-diffusion fronts in the (R T,R c ) plane Classification of the various hydrodynamic instabilities of exothermic reaction-diffusion fronts in the (R T,R c ) plane Double-diffusive instabilities of chemical fronts have some Double-diffusive instabilities of chemical fronts have some differences with pure hydrodynamic DD instabilities: Different base state Different base state stability boundaries depend on the kinetics and on Le stability boundaries depend on the kinetics and on Le different nonlinear dynamics: frozen fingers different nonlinear dynamics: frozen fingers Uncovering of a new instability due to the coupling between thermal diffusion and spatial variations in reaction rate: should be observed in families of exothermic reactions for which  c and  T are both negative Uncovering of a new instability due to the coupling between thermal diffusion and spatial variations in reaction rate: should be observed in families of exothermic reactions for which  c and  T are both negative

25 Take home message When chemical reactions are at the core of density gradients, the possible resulting hydrodynamic instabilities in the corresponding reaction-diffusion-convection system is not always the simple addition of the usual buoyancy related instabilities on a chemical pattern. New chemically-driven instabilities can arise ! References: J. D'Hernoncourt, A. Zebib and A. De Wit, Phys. Rev. Lett. 96, (2006). J. D'Hernoncourt, A. De Wit and A. Zebib, J. Fluid Mech. 576, (2007). J. D'Hernoncourt, A. Zebib and A. De Wit, Chaos, 17, (2007).

26 Front in horizontal set-ups 11  0 ≠  1

27 where Equations

3. Boundary conditions and Marangoni boundary condition at the free surface : (4) with M > 0 : C  M < 0 : C 

Open surface with no buoyancy effects (Ra=0) M = 0 : reaction-diffusion front M = 500 Marangoni effects M = - 500

Asymptotic dynamics : focus on the deformed front surrounded by a stationary asymmetric convection roll M>0 M<0

Closed surface: no Marangoni effects (Ma=0) Ra = 0 : reaction-diffusion front Ra=100:  p  r : products lighter go on top Buoyancy effects Ra= - 100:  p  r : products heavier sink

32 Ra = 100 Ra = -100

Buoyancy effects: Comparison with experiments Experiments in capillary tubes with the Bromate-Sulfite reaction : products heavier than reactants => Ra < 0 Qualitative agreement between experiments and theoretical model A. Keresztessy et al., Travelling Waves in the Iodate-Sulfite and Bromate-Sulfite Systems, J. Phys. Chem. 99, , products reactants

Experiments in capillary tubes with the Iodate-Arsenous Acid reaction : d  /d[I - ] = -1, g/cm 3 M J. Pojman et al., Convective Effects on Chemical Waves, J. Phys. Chem. 95, , V num = 3.24 V num = 4.84 V num = 6.44 Numerical front velocities (10 -3 cm/s) Quantitative agreement between experiments and theoretical model

Asymptotic mixing length Asymmetric Marangoni effects Symmetric buoyancy effects

Constant propagation speed Asymmetric Marangoni effectsSymmetric buoyancy effects

Buoyancy effects : Symmetry between the results for Ra > 0 and Ra < 0 Increase of the front deformation, the propagation speed and the convective motions with Ra and L z Conclusions Marangoni effects : Asymmetry between the results for M > 0 and M < 0 Increase of the front deformation, the propagation speed and the convective motions with M and L z

References Marangoni effects: - - L. Rongy and A. De Wit, ``Steady Marangoni flow traveling with a chemical front", J. Chem. Phys. 124, (2006). - L. Rongy and A. De Wit, ``Marangoni flow around chemical fronts traveling in thin solution layers: influence of the liquid depth", J. Eng. Math. 59, (2007). Buoyancy effects: - - L. Rongy, N. Goyal, E. Meiburg and A. De Wit, ``Buoyancy-driven convection around chemical fronts traveling in covered horizontal solution layers", J. Chem. Phys. 127, (2007).

2008 Gordon conference “Oscillations and dynamic instabilities in chemical systems” July 13-18, 2008 Colby College, Waterville, USA Chair: Vice Chair: Anne De Wit Oliver Steinbock