Hydrodynamic instabilities coupled with complex chemical reactions: control, characterization and their modeling

TESIS DE DOCTORADO HYDRODYNAMIC INSTABILITIES COUPLED WITH COMPLEX CHEMICAL REACTIONS: CONTROL, CHARACTERIZATION, AND THEIR MODELING DARÍO MARTÍN ESCALA VODOPIVEC ESCUELA DE DOCTORADO INTERNACIONAL PROGRAMA DE DOCTORADO EN CIENCIA DE MATERIALES SANTIAGO DE COMPOSTELA AÑO 2020 DECLARACIÓN DEL AUTOR DE LA TESIS HYDRODYNAMIC INSTABILITIES COUPLED WITH COMPLEX CHEMICAL REACTIONS: CONTROL, CHARACTERIZATION, AND THEIR MODELING D. Darío Martín Escala Vodopivec Presento mi tesis, siguiendo el procedimiento adecuado al Reglamento, y declaro que: 1) La tesis abarca los resultados de la elaboración de mi trabajo. 2) En su caso, en la tesis se hace referencia a las colaboraciones que tuvo este trabajo. 3) La tesis es la versión definitiva presentada para su defensa y coincide con la versión enviada en formato electrónico. 4) Confirmo que la tesis no incurre en ningún tipo de plagio de otros autores ni de trabajos presentados por mí para la obtención de otros títulos. En Santiago de Compostela, 29 de diciembre de 2020 Fdo. Darío Martín Escala Vodopivec AUTORIZACIÓN DEL DIRECTOR / TUTOR DE LA TESIS HYDRODYNAMIC INSTABILITIES COUPLED WITH COMPLEX CHEMICAL REACTIONS: CONTROL, CHARACTERIZATION, AND THEIR MODELING D. Alberto Pérez Muñuzuri D. Jorge Carballido-Landeira INFORMAN: Que la presente tesis, corresponde con el trabajo realizado por D. Darío Martín Escala Vodopivec, bajo mi dirección, y a utorizo su presentación , considerando que reúne l os r equisitos exigidos en el R eglamento de Estudios de Doctorado de la USC, y que como director de ésta no incurre en las causas de abstención establecidas en Ley 40/2015. En Santiago de Compostela, 29 de diciembre de 2020 Fdo. Alberto Pérez Muñuzuri Fdo. Jorge Carballido-Landeira As´ ı quedaron terminados los cielos y la tierra, y todo lo que hay en ellos. Al llegar el s´ eptimo d´ ıa, Dios descans´ o porque hab´ ıa terminado la obra que hab´ ıa emprendido. G´ enesis 2:1-2 ...Y en el octavo d´ ıa dio comienzo una tesis... que hoy, afortunadamente, ha llegado a su fin... Thus the heavens and the earth were finished, and all the host of them. And on the seventh day God finished his work that he had done, and he rested on the seventh day from all his work that he had done. Genesis 2:1-2 ...And on the eighth day a thesis began... which fortunately today has come to an end... Quedaron as´ ı rematados o ceo e a terra e todos os elementos. Deus conclu´ ıu no d´ ıa sexto a obra que emprendera, e o s´ etimo d´ ıa repousou de todo o seu traballo. X´ enese 2:1-2 ...E no oitavo d´ ıa deu comezo unha tese... que hoxe, afortunadamente, chegou ao seu fin... Dedicado a Mi Familia, Amigos, Recuerdos y Creencias. Contents Summary ix Resumen xv Resumo xxi Nomenclature and Abbrevations xxvii List of Figures xxix List of Tables xxxv List of publications xxxvii 1 Introduction 1 1.1 Homogeneous Systems and Chemical Reactions . . . . . . . . . . . . . . . . . 1 1.2 TypesofReactions ................................ 3 1.3 TypesofReactors................................. 6 1.4 The Belousov-Zhabotinsky Reaction . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.1 Reduced Kinetic Models of the BZ Reaction . . . . . . . . . . . . . . 8 1.4.2 The 1,4-Cyclohexanedione-Bromate-Acid Oscillatory Reaction . . . . 10 1.5 pH-ShiftingReactions .............................. 12 1.5.1 Inorganic pH-Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.2 Organic pH-Shifting Reactions . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Reaction-Diffusion-Convection (RDC) Systems . . . . . . . . . . . . . . . . . 16 1.6.1 Reaction-Diffusion (RD) Systems . . . . . . . . . . . . . . . . . . . . 17 1.7 Hydrodynamic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.7.1 Flow In Porous Media - Daryc’s Law . . . . . . . . . . . . . . . . . . 21 1.7.2 Hele-ShawCells ............................. 24 1.8 Fingering Instabilities in Hele-Shaw Cells . . . . . . . . . . . . . . . . . . . . 26 1.8.1 Density Fingering Instability . . . . . . . . . . . . . . . . . . . . . . . 26 1.8.2 Viscous Fingering Instability . . . . . . . . . . . . . . . . . . . . . . . 29 1.8.3 Chemically Driven Fingering Instabilities . . . . . . . . . . . . . . . . 31 1.9 ThePoly(AcrylicAcid).............................. 34 1.9.1 Structure and pH-dependence . . . . . . . . . . . . . . . . . . . . . . 35 1.9.2 Chain Length and Concentration Regimes . . . . . . . . . . . . . . . . 36 1.9.3 Ionic Strength Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Contents 1.10 The Shadowgraph and Schlieren Optical Techniques . . . . . . . . . . . . . . 38 1.10.1 Generalities................................ 39 1.10.2 Shadowgraph Technique . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.10.3 Schlieren Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Part I: Density Fingering Instability Driven by the BZ-CHD Oscillator 45 Part I Motivation 45 2 Experimental and Numerical Methods 47 2.1 ExperimentalMethods .............................. 47 2.1.1 Hele-Shaw Cell Construction . . . . . . . . . . . . . . . . . . . . . . 47 2.1.2 InjectionProtocol............................. 47 2.1.3 Optical Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.4 Chemical Recipes and Experimental Designs . . . . . . . . . . . . . . 48 2.1.5 Spectroscopy Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.1.6 Precipitate Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.2 NumericalMethods................................ 55 2.2.1 The BZ-CHD Reaction Chemical Models . . . . . . . . . . . . . . . . 55 2.2.2 Batch System Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2.3 2D Reaction-Diffusion-Convection Model . . . . . . . . . . . . . . . . 57 3 Experimental Results 61 3.1 General System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Descriptive Analysis of the Effect of ∆ρand ε.................. 62 3.2.1 Measuring Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Coupled ∆ρand εVariation by Changing [BrO3–]0............... 65 3.3.1 Measuring Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 Reaction-Diffusion-Convection Interplay . . . . . . . . . . . . . . . . . . . . . 69 3.5 Effect of Varying [CHD]0............................. 70 3.6 ChapterDiscussion ................................ 70 4 Detailed Chemical Analysis 73 4.1 Hele-Shaw Cell Control Experiments . . . . . . . . . . . . . . . . . . . . . . 73 4.1.1 Catalyst Inhibition by NaCl . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.2 Experiments Without CHD . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 UV-VisSpectroscopy............................... 75 4.3 PrecipitateFormation............................... 77 4.4 Nuclear Magnetic Resonance (NMR) Spectroscopy . . . . . . . . . . . . . . . 80 4.5 ChapterDiscussion ................................ 82 5 Numerical Results 83 5.1 Equivalence between Experiments and Reaction Models . . . . . . . . . . . . 83 5.1.1 Qualitative Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1.2 Quantitative Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.1.3 Precipitate Formation Prediction . . . . . . . . . . . . . . . . . . . . . 86 ii DAR´ IO MART´ IN ESCALA VODOPIVEC 5.1.4 Equivalence between Full and Skeleton Models in Batch System . . . . 86 5.2 ModelModification................................ 88 5.3 Reaction-Diffusion-Convection (RDC) Numerical Model . . . . . . . . . . . . 89 5.4 Non-Linear RDC Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4.1 Descriptive Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4.2 Instability Variation as a Function of ∆ρand ε............. 91 5.4.3 Numerical Measuring Observables . . . . . . . . . . . . . . . . . . . . 92 5.5 ChapterDiscussion ................................ 94 Part I Conclusions 97 Part II - Viscous Fingering Instability Driven by pH-Shifting Reactions101 Part II Motivation 101 6 Experimental and Numerical Methods 103 6.1 ExperimentalMethods .............................. 103 6.1.1 Chemical Recipes and Experimental Designs for Dynamic Measurements103 6.1.2 Rheological and pH Measurements . . . . . . . . . . . . . . . . . . . 105 6.1.3 Radial Hele-Shaw Cell Experiments . . . . . . . . . . . . . . . . . . . 105 6.1.4 Instability Setup and Protocol . . . . . . . . . . . . . . . . . . . . . . 106 6.1.5 Circularity Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2 NumericalMethods................................ 108 6.2.1 The FS-PAA Reaction Model . . . . . . . . . . . . . . . . . . . . . . 108 6.2.2 The FSG-PAA Reaction Model . . . . . . . . . . . . . . . . . . . . . 110 6.2.3 Batch System Simulations . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2.4 Reaction-Diffusion (RD) Model Simulations . . . . . . . . . . . . . . 111 6.2.5 Diffusion-Convection (DC) Model Simulations . . . . . . . . . . . . . 113 6.2.6 Mesh Independence Study For The DC Model . . . . . . . . . . . . . 114 7 pH - Viscosity Coupling 115 7.1 CouplingMechanism............................... 115 7.2 SystemDynamics................................. 116 7.3 FS-PAA System Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.3.1 [PAA]0Variation............................. 117 7.3.2 [SO32– ]0Variation............................ 118 7.3.3 [Formaldehyde]0Variation........................ 119 7.4 FSG-PAA System Characterization . . . . . . . . . . . . . . . . . . . . . . . . 119 7.4.1 [GLN]0Variation............................. 119 7.5 The FS-PAA Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.5.1 The Effect of Using HSO32– andNaOH................. 121 7.5.2 The Effect of Using a Short-Chain PAA Molecule . . . . . . . . . . . . 122 7.6 OverlapConcentration .............................. 122 7.7 ChapterDiscussion ................................ 123 iii Contents 8 Viscous Fingering Induced by the FS-PAA Reaction 125 8.1 General System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.2 DescriptiveAnalysis ............................... 126 8.2.1 FlowRateEffect ............................. 127 8.2.2 [Formaldehyde]0Variation in the Displaced Solution . . . . . . . . . . 127 8.2.3 [SO32– ]0Variation in the Displacing Solution . . . . . . . . . . . . . . 127 8.3 QuantitativeAnalysis............................... 128 8.3.1 Interface Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.3.2 Circularity Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.4 Instability Mechanism - Schlieren Experiments . . . . . . . . . . . . . . . . . 131 8.5 Experimental Damkh¨ olerNumbers........................ 133 8.6 ChapterDiscussion ................................ 134 9 Numerical Results 135 9.1 Batch Systems Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.1.1 FS-PAASystem ............................. 135 9.1.2 FSG-PAASystem ............................ 137 9.2 Non-Linear DC Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.3 InterfaceThickness................................ 138 9.4 CircularityVariation ............................... 139 9.4.1 Dependence on [Formaldehyde]0.................... 139 9.4.2 Dependence on [SO32 – ]0......................... 140 9.5 Effects of the Reagent Concentration on the Spatial Viscosity Profiles . . . . . 141 9.6 Effect of Diffusion on the System Behavior . . . . . . . . . . . . . . . . . . . 142 9.7 ChapterDiscussion ................................ 143 Part II Conclusions 145 Part III: Complex Pattern Formation and Viscous Fingering Stabilization 149 Part III Motivation 149 10 Experimental and Numerical Methods 151 10.1ExperimentalMethods .............................. 151 10.1.1 ChemicalRecipes............................. 151 10.1.2 Experimental Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 10.1.3 Radial Hele-Shaw Cell: Experiments and Protocols . . . . . . . . . . . 152 10.1.4 SchlierenImaging ............................ 153 10.1.5 Control Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 10.1.6 Precipitation ............................... 154 10.1.7 Scanning Electron Microscopy (SEM) . . . . . . . . . . . . . . . . . . 155 10.2NumericalMethods................................ 155 10.2.1 Non-Linear Reaction-Diffusion-Convection (RDC) Simulations . . . . 155 10.2.2 Mesh Independence Study For The RDC Model . . . . . . . . . . . . . 157 iv DAR´ IO MART´ IN ESCALA VODOPIVEC 11 Experimental Results 159 11.1CaseI ....................................... 159 11.1.1 General System Overview . . . . . . . . . . . . . . . . . . . . . . . . 159 11.1.2 Descriptive Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 11.1.3 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 11.2CaseII....................................... 164 11.2.1 General System Overview . . . . . . . . . . . . . . . . . . . . . . . . 164 11.2.2 Descriptive Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 11.2.3 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 11.3ChapterDiscussion ................................ 168 12 Chemical Analysis and Instability Mechanism 169 12.1 Experimental Analysis - Control Experiments . . . . . . . . . . . . . . . . . . 169 12.1.1 C1: Influence of the Color Indicator . . . . . . . . . . . . . . . . . . . 169 12.1.2 C2: Influence of the Formaldehyde . . . . . . . . . . . . . . . . . . . 169 12.1.3 C3: Influence of the SO32 – ........................ 170 12.1.4 C4: Influence of the PAA . . . . . . . . . . . . . . . . . . . . . . . . . 170 12.1.5 C5: Chemical Interaction at the Interface . . . . . . . . . . . . . . . . 171 12.2 Physical and Chemical Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 174 12.2.1 CrustFormation ............................. 174 12.2.2 ReactiveFront .............................. 177 12.2.3 Reaction Front Velocity and Damhk¨ oler Number Calculation . . . . . . 178 12.2.4 P´ eclet-Damhk¨ oler Number (PeDa).................... 179 12.3ChapterDiscussion ................................ 181 13 Numerical Model and Results 183 13.1 Reaction-Diffusion-Convection (RDC) Numerical Model . . . . . . . . . . . . 183 13.2 Numerical Results for Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 13.2.1 SystemDynamics ............................ 185 13.2.2 Effect of the Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 185 13.2.3 Circularity Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 185 13.3 Numerical Results for Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 13.3.1 SystemDynamics ............................ 187 13.3.2 Effect of the Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 188 13.3.3 Circularity Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 188 13.3.4 DisplacedVolume ............................ 189 13.4SupplementaryResults .............................. 190 13.4.1 Effect of the Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 190 13.4.2 Effect of Varying Kf........................... 190 13.5ChapterDiscussion ................................ 192 Part III Conclusions 195 General Conclusions 197 Appendices 201 v Contents Appendix A: Preparation of Stock Solutions 201 A.1 Stock Solutions Used For Part I . . . . . . . . . . . . . . . . . . . . . . . . . . 201 A.1.1 CHDSolution .............................. 201 A.1.2 Sodium Bromate Solution . . . . . . . . . . . . . . . . . . . . . . . . 201 A.1.3 FerroinSolution ............................. 201 A.1.4 Sodium Sulfate Solution . . . . . . . . . . . . . . . . . . . . . . . . . 201 A.1.5 Sulfuric Acid Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 202 A.1.6 Sodium Chloride Solution . . . . . . . . . . . . . . . . . . . . . . . . 202 A.1.7 Reactive Mixture: Solutions 1 and 2 . . . . . . . . . . . . . . . . . . . 202 A.1.8 Protocol to prepare the BZ-Agarose Gels . . . . . . . . . . . . . . . . 202 A.2 Stock Solutions Used for Part II . . . . . . . . . . . . . . . . . . . . . . . . . 203 A.2.1 Formaldehyde Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 203 A.2.2 Sodium Sulfite Solution . . . . . . . . . . . . . . . . . . . . . . . . . 203 A.2.3 Poly(Acrylic Acid) [PAA] solutions . . . . . . . . . . . . . . . . . . . 203 A.2.4 Sodium Bisulfite Solution . . . . . . . . . . . . . . . . . . . . . . . . 203 A.2.5 Sodium Hydroxide Solution . . . . . . . . . . . . . . . . . . . . . . . 203 A.2.6 Gluconolactone Solution . . . . . . . . . . . . . . . . . . . . . . . . . 203 A.2.7 Bromothymol Blue Indicator . . . . . . . . . . . . . . . . . . . . . . . 203 A.2.8 Displacing Solution Mixture Preparation . . . . . . . . . . . . . . . . 204 A.3 Stock Solutions Used for Part III . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.3.1 Formaldehyde Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.3.2 Sodium Sulfite Solution . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.3.3 Poly(Acrylic Acid) [PAA] solution . . . . . . . . . . . . . . . . . . . . 204 A.3.4 Sodium Carbonate Solution . . . . . . . . . . . . . . . . . . . . . . . 204 A.3.5 Bromothymol Blue Indicator . . . . . . . . . . . . . . . . . . . . . . . 204 A.3.6 Gluconic Acid Solution (Solution B) . . . . . . . . . . . . . . . . . . . 204 A.3.7 Solution A Mixture Preparation . . . . . . . . . . . . . . . . . . . . . 204 Appendix B: Reaction-Diffusion Systems 205 B.1 MaterialsandMethods .............................. 205 B.1.1 CapillarySystem............................. 205 B.1.2 Non-Convective Agarose-Based System . . . . . . . . . . . . . . . . . 206 B.2 NumericalModels ................................ 207 B.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 B.2.2 1D-RD System Model Setup . . . . . . . . . . . . . . . . . . . . . . . 208 B.2.3 2D-RD System Model Setup . . . . . . . . . . . . . . . . . . . . . . . 208 B.3 ExperimentalResults............................... 209 B.3.1 1D Capillary System . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 B.3.2 Non-Convective Agarose-Based System . . . . . . . . . . . . . . . . . 209 B.4 NumericalResults................................. 212 B.4.1 1.5D-RD Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 B.4.2 2D-RDSimulations............................ 212 B.5 Discussion..................................... 216 vi DAR´ IO MART´ IN ESCALA VODOPIVEC Appendix C: Image Analysis Techniques 217 C.1 Analysis Methods Used for Part I . . . . . . . . . . . . . . . . . . . . . . . . . 217 C.1.1 Calculation of Measuring Observables . . . . . . . . . . . . . . . . . . 217 C.2 Analysis Methods Used for Part II . . . . . . . . . . . . . . . . . . . . . . . . 218 C.2.1 Circularity Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 218 C.2.2 Density Area Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 219 C.3 Analysis Methods Used for Part III . . . . . . . . . . . . . . . . . . . . . . . . 220 C.3.1 Time-Dependent Circularity Calculation . . . . . . . . . . . . . . . . . 220 C.3.2 Average Displacing Profile . . . . . . . . . . . . . . . . . . . . . . . . 220 Appendix D: Supplementary Results 223 D.1 ResultsforPartI ................................. 223 D.1.1 Solutal Expansion Coefficient Calculation . . . . . . . . . . . . . . . . 223 D.1.2 RDC Model Permeability Variation . . . . . . . . . . . . . . . . . . . 223 D.1.3 RDC Control Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 223 D.2 ResultsforPartII................................. 226 D.2.1 Effect of the Color Indicator . . . . . . . . . . . . . . . . . . . . . . . 226 D.2.2 Elasticity Effects and Shear Rate Estimation . . . . . . . . . . . . . . . 226 D.3 ResultsforPartIII ................................ 228 D.3.1 Shear Rate Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 228 D.3.2 P´ ecletNumber(Pe)............................ 228 Appendix E: Copyright Permissions 231 Bibliography 235 vii Summary This thesis presents the work realized in the Non-Linear Physics Group of the University of Santiago de Compostela. This work introduces an interdisciplinary study of systems created from coupling hydrodynamic instabilities in Hele-Shaw cells and complex chemical reactions. Objective and State of the Art The main objective of this thesis is to find a nexus between two different fields: On the one hand, the hydrodynamic instabilities, and on the other hand, the extensive world of complex chemical reactions. Inside the field of hydrodynamics, this work will study a more specific type of instabilities known as fingering instabilities. These instabilities occur when a fluid with low mobility is displaced by a fluid with high mobility [45, 44, 93, 168]. The mobility can be affected by the viscosity, density, surface tension, or temperature of the fluid. The main characteristic of these instabilities is that they exhibit well-defined patterns similar to fingers, that occur during the fluid displacement [45]. The fingering phenomenon was observed in many fields of science and industry. Some examples are: • Enhanced Oil Recovery (EOR). In this case, fingering occurs during the oil extraction process, when water is used to displace the more viscous oil. This phenomenon directly affects the efficiency of the displacement and it is detrimental to the recovery process [132, 206, 41, 21, 207, 210]. • Hydrogeological phenomena. Many studies predict the occurrence of saline fingers within the transport of subsurface aquifers. These types of fingers are observed when the colder and more saline seawater makes contact with fresh water. The differences in salinity and temperature produce a characteristic fingering known as salt fingers. This process favors nutrient oxidation in freshwater transport [111, 200, 171, 76]. • In chromatographic processes. In this separation technique, the components of a mixture are separated when the solution passes through a porous matrix. It has been demonstrated the occurrence of viscous fingering when a viscous sample is eluted using a less viscous solvent. Similar to the EOR case, this process is detrimental for the extraction [173, 32, 42, 198, 166]. However, this work will be more focused on those instabilities produced in experimental devices known as Hele-Shaw cells [89]. These cells are made of two parallel plates separated by a very small gap. This device allows studying the fingering phenomenon in a controlled and Resumen • En los procesos cromatogr´ aficos. La cromatograf´ ıa de exclusi´ on molecular es una t´ ecnica de separaci´ on ampliamente utilizada en la qu´ ımica, biolog´ ıa molecular, biotecnolog´ ıa, entre otros. Es un proceso mediante el cual los solutos de una mezcla son separados por diferencias de tama˜ no, al pasar la misma por un medio poroso. Se ha demostrado, que al querer eluir una muestra viscosa con un solvente menos viscoso se produce digitaci´ on viscosa que afecta negativamente tanto a la extracci´ on como a los rendimientos [173, 32, 42, 198, 166]. Sin embargo, este trabajo se enfoca en el estudio de las inestabilidades producidas en dispositivos experimentales conocidos como celdas de Hele-Shaw [89]. Estos dispositivos est´ an conformadas por dos placas separadas estrechamente una de la otra, permitiendo el estudio del proceso de digitaci´ on de una manera relativamente sencilla y controlada. Poseen adem´ as la particularidad de que el flujo dentro de ellas es id´ entico al de un fluido a trav´ es de un medio poroso, por lo que los estudios llevados a cabo en ellas pueden ser representativos de sistemas muchos m´ as complejos y de particular inter´ es en la industria. El fen´ omeno de digitaci´ on en celdas de Hele-Shaw fue y sigue siendo extensamente estudiado y caracterizado desde hace d´ ecadas . Los primeros estudios abordaron la problem´ atica de desplazamientos no reactivos tanto miscibles como inmiscibles, demostrando la importancia de factores como la tensi´ on superficial o la difusividad en la desestabilizaci´ on de sistemas [93, 48, 127, 197, 8, 45]. M´ as recientemente, se estudiaron los procesos mediados por reacciones qu´ ımicas, en donde sistemas con movilidades iniciales favorables (y por lo tanto, estables), son desestabilizados por procesos fisicoqu´ ımicos acoplados al desplazamiento [45, 25, 44, 9]. El estudio de reacciones qu´ ımicas acopladas al fen´ omeno de digitaci´ on fue incrementando su complejidad con el tiempo. As´ ı, reacciones elementales como procesos de neutralizaci´ on o de precipitaci´ on qu´ ımica, fueron y son hoy en d´ ıa extensamente caracterizados mostrando un rico abanico de comportamientos. El efecto de estas reacciones en las inestabilidades est´ a mediado principalmente por el cambio en las razones de movilidad debido al incremento de factores como la densidad o la viscosidad de los fluidos involucrados o la permeabilidad del medio [141, 138, 137, 139, 73, 65, 9, 176]. La complejidad de los procesos qu´ ımicos acoplados al desarrollo de inestabilidades puede incrementarse considerando lo que denominamos reacciones qu´ ımicas complejas. En un contexto general, estas reacciones son definidas como procesos qu´ ımicos que se producen en varias etapas (al menos dos) y cuentan con al menos un intermediario de reacci´ on [116]. Dentro de esta definici´ on, se engloba un campo muy extenso de reacciones. Algunas de estas, ya han sido acopladas a inestabilidades con anterioridad, como por ejemplo los procesos autocatal´ ıticos [101, 49, 96]. En el contexto de este trabajo, utilizaremos reacciones qu´ ımicas cuyos mecanismos aportan un paso m´ as de complejidad respecto a las reacciones autocatal´ ıticas y donde los intermediarios de reacci´ on desempe˜ nan un papel fundamental para el desarrollo de la din´ amica de las mismas. As´ ı pues, en este trabajo, se analizar´ a el acople de las inestabilidades hidrodin´ amicas con dos tipos de reacciones complejas: La reacci´ on oscilante de Belousov-Zhabotinsky (BZ) [18] y las reacciones de cambio de pH [106, 108, 107, 159, 160, 145]. La reacci´ on BZ, es uno de los osciladores qu´ ımicos m´ as extensamente conocidos. Es un proceso redox en donde los intermediarios de reacci´ on oscilan en el tiempo mediante la din´ amica de activador/inhibidor [82]. Estas oscilaciones se aprecian con cambios de color peri´ odicos en el caso de un sistema perfectamente agitado o con la formaci´ on de patrones xvi DAR´ IO MART´ IN ESCALA VODOPIVEC espaciotemporales en sistemas espacialmente extendidos. Existen variaciones de esta reacci´ on. En este trabajo en particular se opt´ o por usar una reacci´ on conocida como reacci´ on BZ-CHD, en donde el sustrato org´ anico de la receta original se reemplaza por el componente CHD (1,4-Ciclohexanodiona) [113, 112, 187, 188, 190, 114]. Esta formulaci´ on presenta varias ventajas respecto a la receta original, pero la m´ as significativa es que no produce di´ oxido de carbono como producto secundario, lo cual es fundamental si se pretende acoplar esta reacci´ on a fen´ omenos hidrodin´ amicos en medios confinados. Por otra parte, las reacciones de cambio de pH son reacciones en donde el pH del sistema cambia de un estado b´ asico (o ´ acido) a un estado ´ acido (o b´ asico) debido a la interacci´ on de los intermediarios de reacci´ on [107, 108, 159, 153, 145]. Existen reacciones de cambio de pH inorg´ anicas, las cuales son tambi´ en procesos redox y pueden mostrar oscilaciones tanto en sistemas abiertos como en sistemas semi abiertos. Estos sistemas han sido extensamente caracterizados habiendo una gran lista de formulaciones disponibles [145]. Sin embargo, debido a su naturaleza redox, estos sistemas no pueden acoplarse a elementos org´ anicos, como son los pol´ ımeros o sistemas de liberaci´ on de f´ armacos (o drug delivery por su nombre de ingl´ es). Recientemente, se desarrollaron reacciones de cambio de pH org´ anicas que a diferencia de las anteriores no son procesos redox, y por lo tanto son menos agresivas con elementos org´ anicos como los antes indicados [106, 108, 107]. Estas reacciones tambi´ en muestran comportamientos oscilatorios en sistemas abiertos y han sido relativamente bien caracterizadas. Sin embargo, debido a no ser reacciones del tipo redox, hay pocas formulaciones disponibles. En este trabajo se utilizar´ an dos reacciones de cambio de pH org´ anicas, la reacci´ on de Formaldehido-Sulfito (FS) [212, 106, 213] y la reacci´ on de Formaldeido-Sulfito-Gluconolactona (FSG) [108, 107]. La primera es tambi´ en conocida como reacci´ on Clock, porque el pH de la soluci´ on cambia en un momento determinado de ´ acido a b´ asico. El tiempo de cambio depende de las concentraciones de las especies involucradas. La segunda reacci´ on, es una derivaci´ on de la primera en donde se acopla la hidr´ olisis de la gluconolactona al cambio de pH de la reacci´ on de Sulfito-Formaldehido. As´ ı pues, el objetivo de estudio de esta tesis es acoplar entre si todos los fen´ omenos descritos anteriormente. De manera espec´ ıfica se estudiar´ an los siguientes sistemas: •Sistema 1: Acople entre la reacci´ on BZ-CHD y una inestabilidad de digitaci´ on inducida por efecto de la flotabilidad en una celda orientada verticalmente. •Sistema 2: Acople entre la reacci´ on FS y una inestabilidad de digitaci´ on viscosa en una celda orientada horizontalmente. •Sistema 3: Acople entre la reacci´ on FSG y una inestabilidad de digitaci´ on viscosa en una celda orientada horizontalmente. De esta manera se pretende construir estos sistemas experimentalmente y trabajar con ellos en un rango de par´ ametros en donde los tiempos caracter´ ısticos sean comparables y ocurran sinergia entre ambos fen´ omenos. Se realizar´ a la completa caracterizaci´ on, se plantear´ an hip´ otesis respecto a los mecanismos de interacci´ on involucrados y se establecer´ an modelos matem´ aticos que permitan simular la din´ amica de los mismos. xvii Resumen Estructura de la tesis Esta trabajo est´ a dividido en tres partes. Cada parte cuenta con una introducci´ on, un cap´ ıtulo de m´ etodos experimentales y num´ ericos, un cap´ ıtulo de resultados experimentales, un cap´ ıtulo de resultados num´ ericos y una discusi´ on general. Independientemente del hilo conductor com´ un que poseen las partes, se eligi´ o organizar el trabajo de esta forma no solo para facilitar su lectura, sino adem´ as porque cada parte en s´ ı posee suficiente contenido independiente que justifica el uso del formato. De esta manera, los Cap´ ıtulos 2,6y10, son cap´ ıtulos que desarrollan los m´ etodos tanto experimentales como num´ ericos de las Partes I,II yIII respectivamente. En la secci´ on experimental de estos cap´ ıtulos se encuentran todas las recetas, formulaciones qu´ ımicas y protocolos utilizados en los experimentos, as´ ı como la descripci´ on de los arreglos experimentales y estudios previos necesarios para el desarrollo del trabajo. En la secci´ on num´ erica, se incluyen consideraciones generales, como la descripci´ on de los dominios computacionales, configuraci´ on de los programas empleados en las simulaciones, par´ ametros utilizados, entre otros aspectos. Sin embargo y para facilitar la lectura de la tesis, el desarrollo de las ecuaciones fundamentales de los modelos se reserva para los correspondientes cap´ ıtulos de resultados num´ ericos, ya que en la mayor´ ıa de los casos, ´ estos dependen de los resultados experimentales. Los conceptos b´ asicos son introducidos en el Cap´ ıtulo 1. En este cap´ ıtulo se explican una serie de cuestiones fundamentales para el entendimiento del trabajo. As´ ı, se describen las reacciones oscilantes cl´ asicas, como la reacci´ on de Belousov-Zhabotinsky y los osciladores de pH tanto inorg´ anicos como los org´ anicos. Tambi´ en se introducen las inestabilidades hidrodin´ amicas en celda Hele-Shaw y los fen´ omenos de digitaci´ on por diferencia de viscosidad y de densidad. Complementariamente, se incluye una secci´ on para describir las propiedades m´ as importantes del ´ acido poliacr´ ılico, que es un componente ampliamente utilizado en el desarrollo de este trabajo, m´ as precisamente en las Partes II yIII. Finalmente, se describe la t´ ecnica ´ optica de Schlieren. Esta es una t´ ecnica de visualizaci´ on de fluidos ampliamente utilizada durante todo el desarrollo de la tesis y que fue fundamental para el entendimiento de los mecanismos f´ ısicos y qu´ ımicos de los fen´ omenos observados experimentalmente. La Parte I comienza inmediatamente despu´ es del cap´ ıtulo introductorio. En esta parte se estudia el acople entre la reacci´ on BZ-CHD (una variaci´ on de la reacci´ on Belousov-Zhabotinsky), y la inestabilidad hidrodin´ amica de digitaci´ on inducida por flotabilidad en una celda Hele-Shaw orientada verticalmente. En el Cap´ ıtulo 3 se presentan los resultados experimentales obtenidos para este sistema. Los mismos son analizados de forma tanto descriptiva como cuantitativa, estudiando los efectos producidos por los cambios en la densidad (∆ρ) y la excitabilidad (ε). Estos par´ ametros fueron modificados tanto de manera desacoplada como acoplada, y su efecto fue caracterizado mediante el estudio de observables espec´ ıficos como los tiempos de inducci´ on qu´ ımicos (tind−C) e hidrodin´ amicos (tind−H), la longitud de onda (λH) y el per´ ıodo (TC). De esta forma, se presenta un estudio de caracterizaci´ on completo de los fen´ omenos observados, pudi´ endose establecer las principales influencias de la hidrodin´ amica y la qu´ ımica en la din´ amica del sistema. En el Cap´ ıtulo 4 se presenta un estudio qu´ ımico detallado realizado con la finalidad de descubrir el mecanismo detr´ as de los resultados experimentales en el Cap´ ıtulo 3. Dicho estudio se llev´ o a cabo mediante una extensiva serie de experimentos de control y caracterizaci´ on qu´ ımica, en donde se utilizaron diversas herramientas de an´ alisis tales como espectrofotometr´ ıa xviii DAR´ IO MART´ IN ESCALA VODOPIVEC y resonancia magn´ etica nuclear. Mediante estos estudios, es identificado el factor principal en el acople entre la reacci´ on BZ-CHD y la inestabilidad hidrodin´ amica. En el Cap´ ıtulo 5, se aplican los resultados obtenidos en el Cap´ ıtulo 4 en combinaci´ on con los modelos cin´ eticos ya existentes, para reproducir de forma natural los resultados experimentales mediante simulaciones. As´ ı, en este cap´ ıtulo se comparan primeramente los resultados experimentales y los modelos cin´ eticos con la finalidad de obtener un rango de par´ ametros de trabajo equivalente. Luego, se propone un modelo num´ erico extensivo de reacci´ on-difusi´ on-convecci´ on incluyendo los mecanismos descubiertos y desarrollando paso a paso todas las ecuaciones fundamentales. Finalmente, los resultados num´ ericos son analizados an´ alogamente a los experimentales demostrando las equivalencias entre ambos sistemas. La Parte II comienza inmediatamente luego del Cap´ ıtulo 5. En este caso se estudia el acople entre una inestabilidad hidrodin´ amica de digitaci´ on viscosa y una reacci´ on de cambio de pH en una celda de Hele-Shaw orientada horizontalmente. Previamente al estudio del sistema hidrodin´ amico, es necesario desarrollar un sistema qu´ ımico en donde la viscosidad sea afectada por cambios en el pH. De esta manera, en el Cap´ ıtulo 7 se desarrolla un sistema en donde se observan experimentalmente variaciones de temporales sin´ ergicas entre el pH y la viscosidad. Esto se logra acoplando un pol´ ımero sensible al pH, (el ´ acido poliacr´ ılico), con dos reacciones de cambio de pH: la reacci´ on de Formaldehido-Sulfito y la reacci´ on de Formaldehido-Sulfito-Gluconolactona. Ambos sistemas son extensamente caracterizados mediante t´ ecnicas reol´ ogicas y anal´ ıticas, midiendo observables como los saltos de viscosidad y pH (∆µy∆pH respectivamente), los cambios en la din´ amica temporal respecto a la formulaci´ on original y los tiempos de inducci´ on del sistema. Los observables fueron estudiados para cada especie qu´ ımica involucrada en las formulaciones utilizadas. Finalmente, la ´ ultima parte de este cap´ ıtulo demuestra el mecanismo de acople mediante experimentos de control y rutas alternativas considerando formulaciones diversas. En el Cap´ ıtulo 8, los desarrollos obtenidos en el cap´ ıtulo anterior son adaptados a una celda de Hele-Shaw para generar, con ´ exito, una inestabilidad de digitaci´ on viscosa. Este sistema se estudia de manera descriptiva analizando los efectos producidos por los par´ ametros m´ as importantes como son el caudal volum´ etrico (Q) y la composici´ on de las formulaciones utilizadas. Seguidamente el sistema se caracteriza cuantitativamente, estudiando la morfolog´ ıa de los patrones observados mediante el c´ alculo de la circularidad (C). Tambi´ en se utiliza la t´ ecnica de Schlieren como complemento para dilucidar el mecanismo de la inestabilidad. El Cap´ ıtulo 9 presenta los resultados num´ ericos de los sistemas presentados en los Cap´ ıtulos 7 y8. En la primera parte, se adaptan los modelos cin´ eticos existentes de las reacciones de pH a los experimentos. Esto se realiza encontrando el conjunto de par´ ametros que mejor ajusta los modelos a los resultados experimentales. Los modelos ajustados son analizados de manera an´ aloga a como se hace en el Cap´ ıtulo 7. Seguidamente, se presentan las ecuaciones fundamentales de un modelo num´ erico de convecci´ on-difusi´ on para simular el sistema introducido en el Cap´ ıtulo 8. Los resultados son analizados midiendo la circularidad num´ erica de la misma forma que se hizo en los experimentos. Se demuestra que tanto los resultados experimentales como los num´ ericos concuerdan m´ as que aceptablemente. Seguidamente comienza la Parte III en donde se estudia el acople entre la reacci´ on de Sulfito-Formaldehido-Gluconolactona y el fen´ omeno de digitaci´ on viscosa en celda de Hele-Shaw orientada horizontalmente. En esta parte se estudian dos situaciones provenientes de un mismo sistema experimental. Estos casos se obtienen intercambiando solamente la soluci´ on desplazante por la desplazada. En el llamado Caso I, se analiza la generaci´ on de patrones xix Resumen complejos en condiciones de aparente estabilidad hidrodin´ amica. En el Caso II se analiza la estabilizaci´ on de digitaci´ on viscosa mediante procesos fisicoqu´ ımicos. Si bien el sistema experimental estudiado en esta parte es derivado de la Parte II, la riqueza, versatilidad y complejidad de los resultados obtenidos han hecho necesario dedicar un espacio aparte para ellos. As´ ı, el Cap´ ıtulo 11 estudia la din´ amica de los dos casos considerados. En ambas situaciones, se realizaron estudios descriptivos seguidos de estudios cuantitativos para casos reactivos y no reactivos. Los estudios cuantitativos se realizaron mayormente midiendo la circularidad, siendo posible demostrar una fuerte dependencia de los resultados con la velocidad de desplazamiento y la composici´ on qu´ ımica de los sistemas. En ambos casos, el caudal volum´ etrico fue elegido como el par´ ametro de an´ alisis fundamental, siendo este variado en un amplio rango de valores y permitiendo la caracterizaci´ on completa del sistema. El Cap´ ıtulo 12 se centra en encontrar un mecanismo que d´ e explicaci´ on a los fen´ omenos observados. Esto se lleva a cabo mediante un extensivo an´ alisis con experimentos de control, en donde las formulaciones qu´ ımicas fueron modificadas minuciosamente para poder entender el efecto producido por cada uno de las especies en el sistema. Bas´ andose en estos resultados, se propone un modelo qu´ ımico y se realizan c´ alculos complementarios para fundamentarlo. Finalmente, en el Cap´ ıtulo 13 se presenta un modelo num´ erico de reacci´ on-difusi´ on-convecci´ on que simula los fen´ omenos descritos en el Cap´ ıtulo 11, en donde se detallan paso por paso el desarrollo de las ecuaciones fundamentales del mismo. Los resultados num´ ericos se analizan una vez m´ as imitando el an´ alisis realizado a los resultados experimentales, demostrando una gran concordancia entre los mismos. El modelo tambi´ en se utiliza como herramienta para explicar el efecto de la difusi´ on en la din´ amica de los casos de estudio. Una vez finalizada la tercera parte, se presenta una serie de ap´ endices con informaci´ on complementaria. En el Ap´ endice A se muestran todas las recetas de las soluciones madres utilizadas para llevar a cabo este trabajo. En las mismas, se detallan referencias, protocolos y cantidades de los qu´ ımicos utilizados. Tambi´ en se detallan las recetas de las soluciones reactivas usadas en cada parte. En el Ap´ endice B, se presentan los resultados, tanto num´ ericos como experimentales, de los sistemas de reacci´ on-difusi´ on derivados de la Parte I. Estos resultados fueron importantes para el desarrollo y el entendimiento previo del mecanismo propuesto en dicha parte. Sin embargo, se ha decidido incluirlo en un ap´ endice como resultados complementarios. En este ap´ endice se estudian sistemas capilares en 1D y con matriz de agarosa en 2D. En el Ap´ endice C se presentan los m´ etodos y protocolos utilizados para el tratamiento de im´ agenes experimentales y num´ ericas. Estos m´ etodos son fundamentales para la obtenci´ on de los valores cuantitativos presentados a lo largo del trabajo. El ap´ endice est´ a dividido en los m´ etodos utilizados en cada parte. En el Ap´ endice D se incluye informaci´ on y c´ alculos complementarios que fueron utilizados para el desarrollo de los cap´ ıtulos principales, pero que no son incluidos directamente en el cuerpo de la tesis debido a que su aporte no es especialmente significativo. Finalmente, el Ap´ endice E muestra los permisos para el uso, reproducci´ on, modificaci´ on y publicaci´ on del material perteneciente a aquellas publicaciones en donde los derechos de autor fueron cedidos. Estos permisos se incluyen para evitar cualquier problema legal referente al plagio o autoplagio de informaci´ on. xx Resumo A presente memoria resume o traballo realizado no Grupo de F´ ısica Non Lineal da USC. Na mesma exponse o estudo tanto experimental como num´ erico de sistemas xerados mediante a combinaci´ on entre as inestabilidades hidrodin´ amicas en c´ elulas de Hele-Shaw e reacci´ ons qu´ ımicas complexas Obxectivo e Estado do Arte O obxectivo desta tese ´ e o de atopar un nexo com´ un entre dous mundos aparentemente moi diferentes. Por unha banda, o campo das inestabilidades hidrodin´ amicas e por outra, o campo do que neste contexto denominamos reacci´ ons qu´ ımicas complexas. Como todo traballo de especializaci´ on, ´ e fundamental definir concretamente os l´ ımites do estudo. Isto ´ e a´ ında m´ ais necesario, tendo en conta que ´ ambolos dous mundos son extremadamente extensos Dentro do campo das inestabilidades hidrodin´ amicas, neste traballo estud´ asense un tipo particular de inestabilidades co˜ necidas como dixitaci´ ons (ou fingering en ingl´ es). Estas inestabilidades obs´ ervanse cando un flu´ ıdo con certas propiedades termodin´ amicas (como a temperatura, viscosidade, densidade ou a tensi´ on superficial), entra en contacto con outro, e despr´ azao [45, 44, 93, 168]. Se as condici´ ons de desprazamento son desfavorables, o arrastre prod´ ucese de maneira non homox´ enea, xerando patr´ ons similares a dedos (de a´ ı o nome de dixitaci´ on) [45]. Estes sistemas son considerados inestables desde o punto de vista da fluidodin´ amica. O fen´ omeno de dixitaci´ on observouse amplamente en moitos campos da ciencia e a industria, as´ ı por nomear alg´ uns poden citarse: • A extracci´ on mellorada de petr´ oleo ou Enhanced Oil Recovery-EOR polas s´ uas siglas en ingl´ es. Neste caso, o fen´ omeno de dixitaci´ on viscosa ocorre ao querer desprazar o petr´ oleo (un flu´ ıdo cunha maior viscosidade) mediante a inxecci´ on de auga a presi´ on (un flu´ ıdo cunha viscosidade menor). O desprazamento inestable afecta directamente o proceso de extracci´ on, facendo que este sexa ineficiente, e por tanto producindo uns rendementos reducidos [132, 206, 41, 21, 207, 210]. • En fen´ omenos hidroxeol´ oxicos: Varios estudos pred´ ın a aparici´ on de dixitaci´ ons salinas no transporte de acu´ ıferos subterr´ aneos. Este tipo de dixitaci´ ons sucede cando a auga do mar, que pos´ ue unha concentraci´ on salina e temperatura caracter´ ıstica, entra en contacto con auga doce. As diferenzas de salinidade e temperatura producen dixitaci´ ons co˜ necidas como dedos de sal (ou salt fingers en ingl´ es) que interfiren negativamente nas descargas de auga doce, favorecendo a oxidaci´ on de nutrientes [111, 200, 171, 76]. Resumo • Nos procesos cromatogr´ aficos. A cromatograf´ ıa de exclusi´ on molecular ´ e unha t´ ecnica de separaci´ on amplamente utilizada na qu´ ımica, biolox´ ıa molecular, biotecnolox´ ıa, entre outros. ´ E un proceso mediante o cal una os solutos dunha mestura son separados por diferenzas de tama˜ no, ao pasar a mesma por un medio poroso. Demostrouse, que ao querer eluir unha mostra viscosa cun solvente menos viscoso prod´ ucese un fen´ omeno de dixitaci´ on viscosa que afecta negativamente tanto ´ a extracci´ on como aos rendementos [173, 32, 42, 198, 166]. Con todo, este traballo enf´ ocase no estudo das inestabilidades producidas en dispositivos experimentais co˜ necidos como c´ elulas de Hele- Shaw [89]. Estes dispositivos est´ an conformadas por d´ uas placas separadas estreitamente una da outra, permitindo o estudo do proceso de dixitaci´ on dunha maneira relativamente sinxela e controlada. Pos´ uen ademais a particularidade de que o fluxo dentro delas ´ e id´ entico ao dun flu´ ıdo a trav´ es dun medio poroso, polo que os estudos levados a cabo nelas poden ser representativos de sistemas moitos m´ ais complexos e de particular interese na industria. O fen´ omeno de dixitaci´ on en c´ elulas de Hele-Shaw foi e segue sendo cumpridamente estudado e caracterizado desde hai d´ ecadas. Os primeiros estudos abordaron a problem´ atica de desprazamentos non reactivos tanto miscibles como inmiscibles, demostrando a importancia de factores como a tensi´ on superficial ou a difusi´ on na desestabilizaci´ on dos sistemas [93, 48, 127, 197, 8, 45]. Mais recentemente, estud´ aronse os procesos mediados por reacci´ ons qu´ ımicas, onde sistemas con mobilidades iniciais favorables (e por tanto, estables), son deestabilizados por procesos fisicoqu´ ımicos axustados ao desprazamento [45, 25, 44, 9]. O estudo de reacci´ ons qu´ ımicas axustadas ao fen´ omeno de dixitaci´ on foi incrementando a s´ ua complexidade co tempo. As´ ı, reacci´ ons elementais como procesos de neutralizaci´ on ou de precipitaci´ on qu´ ımica, foron e son hoxe en d´ ıa cumpridamente caracterizados mostrando un rico abanico de comportamentos. O efecto destas reacci´ ons nas inestabilidades est´ a mediado principalmente polo cambio nas raz´ ons de mobilidade debido ao incremento de factores como a densidade ou a viscosidade dos flu´ ıdos involucrados ou ´ a permeabilidade do medio [141, 138, 137, 139, 73, 65, 9, 176]. A complexidade dos procesos qu´ ımicos axustados ao desenvolvemento de inestabilidades pode incrementarse considerando o que denominamos reacci´ ons qu´ ımicas complexas. Nun contexto xeral, estas reacci´ ons son definidas como procesos qu´ ımicos que se producen en varias etapas (polo menos d´ uas) e contan con polo menos un intermediario de reacci´ on [116]. Dentro desta definici´ on, engl´ obase un campo moi extenso de reacci´ ons. Algunhas destas, xa foron axustadas a inestabilidades con anterioridade, por exemplo os procesos autocatal´ ıticos [101, 49, 96]. No contexto deste traballo, utilizaremos reacci´ ons qu´ ımicas cuxos mecanismos achegan un paso m´ ais de complexidade respecto a as reacci´ ons autocatal´ ıticas e onde os intermediarios de reacci´ on desempe˜ nan un papel fundamental para o desenvolvemento da din´ amica das mesmas. As´ ı pois, neste traballo, analizarase o acoplamento das inestabilidades hidrodin´ amicas con dous tipos de reacci´ ons complexas: A reacci´ on oscilante de Belousov- Zhabotinsky (BZ) [18] e as reacci´ ons de cambio de pH [106, 108, 107, 159, 160, 145]. A reacci´ on BZ, ´ e un dos osciladores qu´ ımicos m´ ais cumpridamente co˜ necidos. ´ E un proceso redox onde os intermediarios de reacci´ on oscilan no tempo mediante a din´ amica de activador/inhibidor [82]. Estas oscilaci´ ons apr´ ecianse con cambios de cor peri´ odicos no caso dun sistema perfectamente axitado ou coa formaci´ on de patr´ ons espaciotemporales en sistemas xxii DAR´ IO MART´ IN ESCALA VODOPIVEC espacialmente estendidos. Existen variaci´ ons desta reacci´ on. Neste traballo en particular optouse por usar unha reacci´ on co˜ necida como reacci´ on BZCHD, onde o substrato org´ anico da receita orixinal substit´ uese polo compo˜ nente CHD (1,4-Ciclohexanodiona) [113, 112, 187, 188, 190, 114]. Esta formulaci´ on presenta varias vantaxes respecto a a receita orixinal, pero a m´ ais significativa ´ e que non produce di´ oxido de carbono como produto secundario, o cal ´ e fundamental se se pretende axustar esta reacci´ on a fen´ omenos hidrodin´ amicos. Por outra banda, as reacci´ ons de cambio de pH son reacci´ ons onde o pH do sistema cambia dun estado alcalino (ou aceda) a un estado acedo (ou alcalino) debido ´ a interacci´ on dos intermediarios de reacci´ on [107, 108, 159, 153, 145]. Existen reacci´ ons de cambio de pH inorg´ anicas, as cales son tam´ en procesos redox e poden mostrar oscilaci´ ons tanto en sistemas abertos como en sistemas semi abertos. Estes sistemas foron cumpridamente caracterizados habendo unha gran lista de formulaci´ ons dispo˜ nibles [145]. Con todo, debido ´ a s´ ua natureza redox, estes sistemas non poden axustarse a elementos org´ anicos, como os pol´ ımeros ou elementos de transporte de f´ armacos (ou drug delivery polo seu nome de ingl´ es). Mais recentemente, desenvolv´ eronse reacci´ ons de cambio de pH org´ anicas que a diferenza das anteriores non son procesos redox, e por tanto son menos agresivas con elementos org´ anicos como os antes indicados [106, 108, 107]. Estas reacci´ ons tam´ en mostran comportamentos oscilatorios en sistemas abertos e foron relativamente caracterizadas. Con todo, debido a non ser reacci´ ons do tipo redox, hai poucas formulaci´ ons dispo˜ nibles. Neste traballo utilizaranse d´ uas reacci´ ons de cambio de pH org´ anicas, a reacci´ on de Formaldehido-Sulfito (FS) [212, 106, 213], e a reacci´ on de Formaldeido-Sulfito-Gluconolactona (FSG) [108, 107]. A primeira ´ e tam´ en co˜ necida como reacci´ on Clock, porque o pH da soluci´ on cambia nun momento determinado de acedo a alcalino. O tempo de cambio depende das concentraci´ ons das especies involucradas. A segunda reacci´ on, ´ e unha derivaci´ on da primeira onde se axusta a hidr´ olise da gluconolactona ao cambio de pH da reacci´ on de Formaldehido-Sulfito. A hidr´ olise converte a gluconolactona en ´ acido gluc´ onico producindo un incremento puntual do pH. As´ ı pois, o obxectivo de estudo desta tese ´ e axustar entre se todos os fen´ omenos descritos anteriormente. De maneira espec´ ıfica estudaranse tres sistemas axustados: •Sistema 1: Acoplamento entre a reacci´ on BZ-CHD e unha inestabilidade de dixitaci´ on inducida por flotabilidade en nunha c´ elula orientada verticalmente. •Sistema 2: Acoplamento entre a reacci´ on FS e unha inestabilidade de dixitaci´ on viscosa nunha c´ elula orientada horizontalmente. •Sistema 3: Acoplamento entre a reacci´ on FSG e unha inestabilidade de dixitaci´ on viscosa nunha c´ elula orientada horizontalmente. Desta maneira pret´ endese constru´ ır estes sistemas experimentalmente, e traballar con eles nun rango de par´ ametros onde os tempos caracter´ ısticos sexan comparables e ocorran sinerxias entre ambos fen´ omenos. Realizarase a completa caracterizaci´ on, exporanse hip´ otese respecto a os mecanismos de interacci´ on involucrados e estableceranse modelos matem´ aticos que permitan simular a din´ amica dos mesmos. xxiii Resumo Estrutura da Tese Esta traballo est´ a dividido en tres partes. Cada parte conta cunha introduci´ on, un cap´ ıtulo de m´ etodos experimentais e num´ ericos, un cap´ ıtulo de resultados experimentais, un cap´ ıtulo de resultados num´ ericos e unha discusi´ on xeral. Independentemente do f´ ıo condutor com´ un que pos´ uen as partes, elixiuse organizar o traballo desta forma non s´ o para facilitar a s´ ua lectura, sen´ on ademais porque cada parte en si pos´ ue suficiente contido independente que xustifica o uso do formato. As´ ı, os Cap´ ıtulos 2,6e10 son cap´ ıtulos que desenvolven tanto o m´ etodo experimental como o num´ erico das Partes I,II eIII respectivamente. Na secci´ on experimental destes cap´ ıtulos atopar´ as todas as receitas, formulaci´ ons qu´ ımicas e protocolos empregados nos experimentos, as´ ı como a descrici´ on dos arranxos experimentais e estudos previos necesarios para o desenvolvemento do traballo. Na secci´ on num´ erica incl´ uense consideraci´ ons xerais, como a descrici´ on dos dominios computacionais, a configuraci´ on dos programas empregados nas simulaci´ ons, par´ ametros empregados, entre outros. Non obstante, e para facilitar a lectura da tese, o desenvolvemento das ecuaci´ ons fundamentais dos modelos res´ ervase para os correspondentes cap´ ıtulos de resultados num´ ericos, xa que na maior´ ıa dos casos, estes dependen dos resultados experimentais. Os conceptos b´ asicos son introducidos no Cap´ ıtulo 1. Neste cap´ ıtulo expl´ ıcanse unha serie de cuesti´ ons fundamentais para o entendemento do traballo. As´ ı, descr´ ıbense as reacci´ ons oscilantes cl´ asicas, como a reacci´ on de Belousov- Zhabotinsky e os osciladores de pH tanto inorg´ anicos como os org´ anicos. Tam´ en se introducen as inestabilidades hidrodin´ amicas en c´ elula Hele-Shaw e os fen´ omenos de dixitaci´ on por diferenza de viscosidade e de densidade. Complementariamente, incl´ uese unha secci´ on para describir as propiedades m´ ais importantes do ´ acido poliacr´ ılico, que ´ e un compo˜ nente amplamente utilizado no desenvolvemento deste traballo, m´ ais precisamente nas Partes II eIII. Finalmente, descr´ ıbese a t´ ecnica ´ optica de Schlieren. Esta ´ e unha t´ ecnica de visualizaci´ on de flu´ ıdos amplamente utilizada durante todo a tese e que foi fundamental para o entendemento dos mecanismos f´ ısicos e qu´ ımicos observados experimentalmente. AParte I comeza inmediatamente despois do cap´ ıtulo introdutorio. Nesta parte est´ udase o axuste entre a reacci´ on BZ-CHD, e a inestabilidade hidrodin´ amica de dixitaci´ on por inducida pola flotabilidade nunha c´ elula Hele-Shaw orientada verticalmente. No Cap´ ıtulo 3 pres´ entanse os resultados experimentais obtidos para este sistema. Os mesmos son analizados de forma tanto descritiva como cuantitativa, estudando os efectos producidos polos cambios na densidade (∆ρ) e da excitabilidade (ε). Estas par´ ametros foron modificados tanto de maneira independente como axustada, e o seu efecto foi caracterizado mediante o estudo de observables espec´ ıficos como os tempos de induci´ on qu´ ımicos (tind−C) e hidrodin´ amicos (tind−H), a lonxitude de onda (λH) e o per´ ıodo (TC). Desta forma, pres´ entase un estudo de caracterizaci´ on completo dos fen´ omenos observados, pod´ endose establecer as principais influencias da hidrodin´ amica e a qu´ ımica na din´ amica do sistema. No Cap´ ıtulo 4 pres´ entase un estudo qu´ ımico detallado realizado coa finalidade de descubrir o mecanismo detr´ as dos resultados experimentais no Cap´ ıtulo 3. Devandito estudo levou a cabo mediante unha extensiva serie de experimentos de control e caracterizaci´ on qu´ ımica, onde se utilizaron diversas ferramentas de an´ alises tales como espectrofotometr´ ıa e resonancia magn´ etica nuclear. Mediante estes estudos, ´ e identificado o factor principal no acoplamento entre a reacci´ on BZ-CHD e a inestabilidade hidrodin´ amica. xxiv DAR´ IO MART´ IN ESCALA VODOPIVEC No Cap´ ıtulo 5, apl´ ıcanse os resultados obtidos no Cap´ ıtulo 4 en combinaci´ on coa os modelos cin´ etico xa existentes, para reproducir de forma natural os resultados experimentais mediante simulaci´ ons. As´ ı, neste cap´ ıtulo comp´ aranse primeiramente os resultados experimentais e os modelos cin´ eticos coa finalidade de obter un rango de par´ ametros de traballo equivalente. Logo, proponse un modelo num´ erico extensivo de reacci´ on-difusi´ on-convecci´ on inclu´ ındo os mecanismos descubertos e desenvolvendo paso a paso todas as ecuaci´ ons fundamentais. Finalmente, os resultados num´ ericos son analizados analogamente aos experimentais demostrando as equivalencias entre ´ ambolos sistemas. AParte II comeza inmediatamente a finalizar o Cap´ ıtulo 5. Neste caso est´ udase o axuste entre unha inestabilidade hidrodin´ amica de dixitaci´ on viscosa e unha reacci´ on de cambio de pH nunha c´ elula de Hele-Shaw orientada horizontalmente. Previamente ao estudo do sistema hidrodin´ amico, ´ e necesario desenvolver un sistema qu´ ımico onde a viscosidade sexa afectada por cambios no pH. Desta maneira, no Cap´ ıtulo 7desenv´ olvese un sistema onde se observan experimentalmente variaci´ ons de temporais sin´ ergicas entre o pH e a viscosidade. Isto l´ ograse mediante o acoplamento entre un pol´ ımero sensible ao pH, (o ´ acido poliacr´ ılico), con d´ uas reacci´ ons de cambio de pH: a reacci´ on de Formaldehido-Sulfito e a reacci´ on de Formaldehido-Sulfito-Gluconolactona. ´ Ambolos sistemas son cumpridamente caracterizados mediante t´ ecnicas reol´ oxicas e anal´ ıticas, medindo observables como os saltos de viscosidade e pH (∆pH e ∆µrespectivamente), os cambios na din´ amica temporal respecto a a formulaci´ on orixinal e os tempos de induci´ on do sistema. Os observables foron estudados para cada especie qu´ ımica involucrada nas formulaci´ ons utilizadas. Finalmente, a ´ ultima parte deste cap´ ıtulo demostra o mecanismo de acoplamento mediante experimentos de control e roteiros alternativas considerando formulaci´ ons diversas. No Cap´ ıtulo 8, o os desenvolvementos obtidos no cap´ ıtulo anterior son adaptados a unha c´ elula de Hele-Shaw para xerar, con ´ exito, unha inestabilidade de dixitaci´ on viscosa. Este sistema est´ udase de maneira descritiva analizando os efectos producidos polos par´ ametros m´ ais importantes como son o caudal volum´ etrico (Q) e a composici´ on das formulaci´ ons utilizadas. Seguidamente o sistema caracter´ ızase cuantitativamente, estudando a morfolox´ ıa dos patr´ ons observados mediante o c´ alculo da circularidade (C). Tam´ en se utiliza a t´ ecnica de Schlieren como un complemento para dilucidar o mecanismo da inestabilidade. OCap´ ıtulo 9 presenta os resultados num´ ericos dos sistemas presentados nos Cap´ ıtulos 7e8. Na primeira parte, ad´ aptanse os modelos cin´ eticos existentes das reacci´ ons de pH a experimentos. Isto real´ ızase atopado en conxunto de par´ ametros que mellor axustan os modelos aos resultados experimentais. Os modelos axustados son analizados de maneira an´ aloga a como se fai no Cap´ ıtulo 7. Seguidamente, pres´ entanse as ecuaci´ ons fundamentais dun modelo num´ erico convecci´ on-difusi´ on para simular o sistema introducido no Cap´ ıtulo 8. Os resultados son analizados medindo a circularidade num´ erica da mesma forma que se fixo nos experimentos. Dem´ ostrase que os resultados experimentais e num´ ericos concordan m´ ais aceptablemente. Seguidamente comeza a Parte III onde se estuda o acoplamento entre unha reacci´ on de pH e o fen´ omeno de dixitaci´ on viscosa en c´ elula de Hele-Shaw orientada horizontalmente. A´ ında que parece repetido ´ a parte anterior, aqu´ ı darase un paso m´ ais no incremento de complexidade. Nesta parte est´ udanse d´ uas situaci´ ons provenientes dun mesmo sistema experimental. Estes casos obt´ e˜ nense intercambiado soamente a soluci´ on desplazante pola desprazada. No chamado Caso I, anal´ ızase a xeraci´ on de patr´ ons complexos en condici´ ons de aparente estabilidade hidrodin´ amica. No Caso II anal´ ızase a estabilizaci´ on de dixitaci´ on viscosa mediante procesos xxv List of Figures 8.9 Temporal evolution of the initial condition in a colorless experiment. . . . . . . 132 8.10 Fingering onset of a (a) colored and a (b) colorless experiment. . . . . . . . . . 132 8.11 Variation of the normalized viscosity µ∗...................... 133 9.1 Characterization of the FS-PAA model by varying [Formaldehyde]0. . . . . . . 136 9.2 Characterization of the FS-PAA model by varying [SO32– ]0........... 136 9.3 Characterization of the FSG-PAA model by varying [GLN]0........... 137 9.4 (a) Viscosity field for a control simulation and (b) schematics of the initial viscosity profile across the radial coordinate. . . . . . . . . . . . . . . . . . . . 138 9.5 Initial condition thickness as a function of (a) [Formaldehyde]0and (b) [SO32– ]0.139 9.6 Circularity values obtained from the convective simulations as a function of [Formaldehyde]0andQ. ............................. 140 9.7 Circularity values obtained from the convective simulations as a function of [SO32– ]0forQ=of7mL/min........................... 140 9.8 Comparison between numerical viscosity reaction-diffusion profiles for different (a) [Formaldehyde]0and (b) [SO32– ]0.................. 142 9.9 OH–concentration profiles of the initial condition (blue mixing zone) obtained fromtheRDsystem. ............................... 142 10.1 Schematics of the configuration of the displacing-displaced fluid for each case ofstudy....................................... 152 10.2 Schematics of the experimental setup. . . . . . . . . . . . . . . . . . . . . . . 153 10.3SEMequipment. ................................. 155 10.4 2D-RDC domain used in the non-linear simulations. . . . . . . . . . . . . . . . 156 10.5 Mesh independence study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 11.1 Comparison between (a) reactive and (b) non-reactive cases. . . . . . . . . . . 160 11.2 Example of the pattern dynamic observed at low flow rate. . . . . . . . . . . . 161 11.3 Effect of the flow rate on the pattern formation. . . . . . . . . . . . . . . . . . 161 11.4 Schlieren visualization for an experiment with Q = 3 µL/min. . . . . . . . . . . 162 11.5 Quantitative comparison of the radial profile as a function of Q. . . . . . . . . . 163 11.6 Quantitative comparison of the displacing solution profiles in a relative timescale for (a) reactive and (b) non-reactive cases. . . . . . . . . . . . . . . . 164 11.7 Stabilization of an initially unstable front for (a) Q = 10 µL/min and (b) Q = 200 µL/min..................................... 165 11.8 Reacting interfaces for several flow rates. . . . . . . . . . . . . . . . . . . . . 166 11.9 Case II observed using the Schlieren technique for (a) 5 µL/min, and (b) 500 µL/min....................................... 167 11.10Quantitative measurements of the morphological changes of the interface as a functionoftheflowrateQ............................. 167 11.11Comparison between the displaced volumes for the reactive and control cases for Q = 10 µL/min................................. 168 12.1 C1: Schlieren images for a control experiment without C·I............ 170 12.2 C2: Control experiment where the formaldehyde was removed from the displacingcomposition............................... 171 12.3 C3: Control experiment the the displacing solution was composed only of PAA. 172 xxxii DAR´ IO MART´ IN ESCALA VODOPIVEC 12.4 C4: Control experiment with only sulfite in the displacing solution composition. 173 12.5 Equilibrium of (a) CO2and (b) SO2species in aqueous solution at 23°C . . . . 173 12.6 C5: Control experiment where the displacing solution was composed of PAA and CO32– . .................................... 174 12.7 Precipitation test done for three different preparations. . . . . . . . . . . . . . 175 12.8 SEM micrographs of Solution A and the precipitate. . . . . . . . . . . . . . . . 176 12.9 Closer observation of the front stabilization in the reverse experiment. . . . . . 177 12.10Measurement of the reaction front velocity used to estimate the Damhk¨ oler number (Da). ................................... 179 12.11Damhk¨ oler number (Da) estimation as a function of the average interface radius. 180 12.12PeDa number estimation for different diffusion rates of Solution B. . . . . . . . 181 13.1 Equivalence between the reaction rate and the front velocity used for the numerical simulation of Case I. . . . . . . . . . . . . . . . . . . . . . . . . . . 185 13.2 Numerical results of the simulation of Case I. . . . . . . . . . . . . . . . . . . 186 13.3 Effect of changing the flow rate in the numerical simulation of Case I. . . . . . 186 13.4 Circularity calculation as a function of the flow rate for Case I. . . . . . . . . . 187 13.5 Numerical results of the simulation of Case II. . . . . . . . . . . . . . . . . . . 187 13.6 Closer observation of the precipitation effect in a simulation of Case II . . . . . 188 13.7 Simulation results of the effect of Q on Case II. . . . . . . . . . . . . . . . . . 188 13.8 Circularity calculation as a function of the flow rate for Case II. . . . . . . . . . 189 13.9 Quantitative numerical comparison between the displaced volumes for the reactive and control cases for Q = 10 and µL/min................. 190 13.10Effect of the diffusion coefficient of species B for (a) the circularity and (b) the patternformation.................................. 191 13.11Effect of the variation of Kf=k1/k2........................ 191 13.12Experimental observation of the effect of changing Kf. ............. 192 B.1 Capillary reactor used in 1D reaction-diffusion experiments. . . . . . . . . . . 206 B.2 Schematics of the 2D reaction-diffusion reactor. . . . . . . . . . . . . . . . . . 207 B.3 1.5D numerical domain used to simulate the capillary system. . . . . . . . . . 208 B.4 2D numerical domain used to simulate the RD model. . . . . . . . . . . . . . . 209 B.5 Space-Time plots obtained from the 1D-RD capillary system for different values of ε......................................... 210 B.6 Experimental 2D-RD experiments for different values of ε............ 211 B.7 STPs obtained from the non-convective agarose-based system. . . . . . . . . . 211 B.8 STPs obtained from the ferroin concentration field by simulating the capillary system in a 1.5D numerical domain. . . . . . . . . . . . . . . . . . . . . . . . 213 B.9 2D-RD simulations results obtained from the modified skeleton model for severalexcitabilities. ............................... 214 B.10 STPs obtained from the 2D-RD simulations. . . . . . . . . . . . . . . . . . . . 215 C.1 Method used to calculate (TCand tind−C). .................... 218 C.2 Method used to calculate λHand tind−H...................... 218 C.3 Schematic of the procedure to calculate the circularity. . . . . . . . . . . . . . 219 C.4 (a) Finger density area dAcalculated for Q = 20 mL/min on Figure 8.6. Both results agree with those obtained using the circularity. (b) dAcalculated at fixed distance d = 108 mm as a function of the Formaldehyde]0. ........... 220 xxxiii List of Figures C.5 Circularity calculation for Cases I and II . . . . . . . . . . . . . . . . . . . . . 221 C.6 Procedure to calculate the average displacing profiles for Case I . . . . . . . . 222 D.1 Fitting for the solutal expansion coefficients. . . . . . . . . . . . . . . . . . . . 224 D.2 Permeability as a function of the precipitate concentration. . . . . . . . . . . . 224 D.3 2D RDC simulation without including the generation of [Q·H2Q]. . . . . . . . 225 D.4 Study of the effect produced by the color indicator in the viscosity of the displacingsolution................................. 226 D.5 Measurement of the first normal stress difference N1for long-chain (PAA-4x106 gmol−1) and short-chain (PAA-4.5x106gmol−1) polymers. . . . . . . . . . . 228 D.6 First Normal Stress Difference (N1) measured for Solution A and a high elasticity reference solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 xxxiv List of Tables 2.1 Base BZ-CHD recipe used in the RDC experiments where εand ∆ρvaried independently.................................... 51 2.3 Correspondence between ε-[H2SO4]0and ∆ρ-[Na2SO4]0. ........... 51 2.4 Base BZ-CHD recipe used in the RDC experiments where εand ∆ρvaried coupled by changing [BrO3–]0and [Br–]0. Both reagents were changed simultaneously as indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5 Correspondence between εand ∆ρwith [BrO3–]0. ............... 53 2.6 Recipe used for studying changes in εdue to changes in [H2SO4]0. These experiments were performed in a batch reactor. . . . . . . . . . . . . . . . . . 54 2.7 Values of εcorresponding to changes in [H2SO4]0for the recipe presented in TableRecipe-5. .................................. 54 2.8 Full chemical model of the BZ-CHD reaction. . . . . . . . . . . . . . . . . . . 56 2.9 Skeleton model of the BZ-CHD reaction. . . . . . . . . . . . . . . . . . . . . 57 2.10 Parameters used in the RDC simulations. . . . . . . . . . . . . . . . . . . . . . 59 6.1 Experimental recipes used to study the effect of each specific reagent in the dynamics of the FS-PAA reaction. Concentration values are expressed as wt% for the PAA and molarity for Na2SO3and Formaldehyde. . . . . . . . . . . . . 104 6.2 Recipe used to analyze the effect of the [GLN]0in the FS-PAA system. Concentration values are expressed as wt% for the PAA and molarity for Na2SO3, Formaldehyde, and GLN. . . . . . . . . . . . . . . . . . . . . . . . . 105 6.3 Solution configuration used to study the influence of changes in the displaced fluid in the Hele-Shaw cell experiments. . . . . . . . . . . . . . . . . . . . . . 107 6.4 Solution configuration used to study the influence of changes in the displacing fluid in the Hele-Shaw cell experiments. . . . . . . . . . . . . . . . . . . . . . 107 6.5 Kinetic model of the FS reaction. . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.6 Set of initial conditions used to simulate the FS-PAA system by varying [Formaldehyde]0.................................. 109 6.7 Set of initial conditions used to simulate the FS-PAA system by varying [SO32– ]0.109 6.8 Kinetic model of the FSG-PAA reaction. . . . . . . . . . . . . . . . . . . . . . 110 6.9 Set of initial conditions used to simulate the FSG-PAA system by varying [GLN]0.110 6.10 Set of initial conditions used in the RD simulations to study changes on the thickness of the initial mixing region by varying [Formaldehyde]0........ 112 6.11 Set of initial conditions set used in the RD simulations to study changes on the thickness of the initial mixing region by varying [SO32– ]0. ........... 112 8.1 Experimental Damk¨ ohlernumbers......................... 134 List of Tables 10.1 Composition of Solutions A and B. . . . . . . . . . . . . . . . . . . . . . . . . 152 10.2 Composition of the control experiments. . . . . . . . . . . . . . . . . . . . . . 154 10.3 Parameters used in the RDC simulations. . . . . . . . . . . . . . . . . . . . . . 157 B.1 Recipe used in the non-convective agarose 2D-Reaction Diffusion System. . . . 207 xxxvi List of Publications The following list indicates the publications derived from this research that were used to create the thesis. I declare that I am the main author of all these publications and no other non-doctor collaborator was part of them. I also declare that I am properly authorized to use this articles in this context, and they were not used in any other thesis work. The corresponding authorizations are detailed in Appendix E. (I) D. M. Escala, M. A. Budroni, J. Carballido-Landeira, A. De Wit, and A. P. Mu˜ nuzuri. Self-organized traveling chemo-hydrodynamic fingers triggered by a chemical oscillator. Journal of Physical Chemistry Letters, 5(3):413–418, 2014. Impact Factor (JCR): 7.458 (2014) Quartile: Q1 (II) D. M. Escala, A. P. Mu˜ nuzuri, A. De Wit, and J. Carballido-Landeira. Temporal viscosity modulations driven by a pH sensitive polymer coupled to a pH-changing chemical reaction. Physical Chemistry Chemical Physics, 19(19):11914–11919, 2017. Impact Factor (JCR): 3.906 (2017) Quartile: Q1 (III) D. M. Escala, A. De Wit, J. Carballido-Landeira, and A. P. Munuzuri. Viscous Fingering Induced by a pH-Sensitive Clock Reaction. Langmuir, 35(11):4182–4188, 2019. Impact Factor (JCR): 3.557 (2019) Quartile: Q1 (IV) D. M. Escala and A. P. Mu˜ nuzuri. Interface Fingering Instability Triggered by a Density-Coupled Oscillatory Chemical Reaction via Precipitation. Langmuir, 35(42):13769–13781, 2019. Impact Factor (JCR): 3.557 (2019) Quartile: Q1 (V) D. M. Escala and A. P´ erez-Mu˜ nuzuri. Constructing or Deconstructing a Fluid Instability: A Bottom-Up Approach. Submitted, 2021. Status: Submitted The next lists indicate the publications that are derived from the thesis research, but they were not included as part of the manuscript due to copyright or permission issues: Articles • D. M. Escala, J. Guiu-Souto, and A. P. Mu˜ nuzuri. Externally controlled anisotropy in pattern-forming reaction-diffusion systems. Chaos, 25(6), 2015. List of Publications • C. A. Middleton, C. Thomas, D. M. Escala, J. L. Tison, and A. De Wit. Imaging the Evolution of Brine Transport in Experimentally Grown Quasi-two-dimensional Sea Ice. In Procedia IUTAM, volume 15, pages 95–100, 2015. • M. A. Budroni, L. Lemaigre, D. M. Escala, A. P. Mu˜ nuzuri, and A. De Wit. Spatially Localized Chemical Patterns around an A + B  Oscillator Front. Journal of Physical Chemistry A, 120(6):851–860, 2016. Book Chapters • D. M. Escala, J. Guiu-Souto, J. Carballido-Landeira, A. P´ erez-Mu˜ nuzuri, and M. E. V´ azquez-Cend´ on. Changes in buoyancy-driven instabilities using a reaction-diffusion system. Numerical Methods for Hyperbolic Equations: Theory and Appl., An Int. Conf. to Honour Professor E.F. Toro - Proc. of the Int. Conf. on Numerical Methods for Hyperbolic Equations: Theory and Appl., pages 397–400, 2013. • J. Guiu-Souto, D. M. Escala, J. Carballido-Landeira, A. P´ erez-Mu˜ nuzuri, and E. Mart´ ın-Ortega. Viscous fingering instabilities in reactive miscible media. Numerical Methods for Hyperbolic Equations: Theory and Appl., An Int. Conf. to Honour Professor E.F. Toro - Proc. of the Int. Conf. on Numerical Methods for Hyperbolic Equations: Theory and Appl., 409:409–412, 2013. xxxviii Chapter 1 Introduction Abstract:This chapter will present the most basics concepts related to all subjects addressed in this work. In this way, a general and descriptive review of the most important concepts on every single studied field will be addressed. The topics will be introduced from simplicity to complexity, emphasizing all that is necessary to understand the experimental and theoretical developments. 1.1 Homogeneous Systems and Chemical Reactions From the point of view of an active environment, a homogeneous system is defined as a material system in which all its intensive properties (such as density, elasticity, temperature, pressure, etc) are constant in the medium. In other words, its composition is uniform for every point of the system. Thus, a perfectly mixed dissolution of salt in water or a block of iron can be considered homogeneous systems. These types of systems can be classified is five different categories [116, 13]: •Dissolutions: Systems composed of only one single phase. These types of systems are composed of at least, one solvent and one solute. •Pure substances: Systems composed of only one substance. •Open systems: Systems where mass and energy are exchanged from the medium to the environment. •Closed systems: Systems where only energy is exchanged between the medium and the environment, but not mass. •Isolated systems: Systems where neither mass nor energy are exchanged with the medium. Due to its homogeneous character, these types of systems can be represented by a zero spatial dimension model. Regardless of the intrinsic complexity of every individual case, most active mediums can be studied in a deterministic manner by using differential equations. Thus, the state and dynamics of the system can be described by defining a set of time-dependent variables Ci(t) = (C1(t),C2(t),...,Cn(t)), and a coupled system of differential equations as: Chapter 1. Introduction dCi(t) dt =fi(Ci(t),pi)(1.1) where fidescribes the dynamics of the system and depends on a set of parameters pi= (p1,p2,...,pm). The expressions of fand pdepend on the system of study [116]. The set of Eqs. (1.1) are often used to describe the kinetics of chemical reactions. In this case, the dynamics are described by the rates of reaction of the species involved. Considering the following homogeneous reaction: aA +bB +··· → eE +f F +... (1.2) where a,b,...,e,f,... are the stoichiometric coefficients, and A,B,...,E,F,... are the chemical species. The rate of consumption or conversion rate (J) of each reactant is proportional to their correspondent stoichiometric coefficient (if the reaction occurs in a closed system), thus: J≡ −1 a dnA dt =−1 b dnB dt =··· =−1 e dnE dt =−1 f dnF dt =... (1.3) where niare the moles of the species i=A,B,.... The conversion rate, J, is an extensive property that depends on the volume of the system. The conversion rate per volume unit J/Vis defined as the reaction rate r[116, 13]: r≡J V=1 V−1 a dnA dt (1.4) ris an intensive magnitude and depends on the temperature (T), the pressure (p), and the concentration of the species [116]. If the volume of the system remains constant, then: 1 V dnA dt =dnA/V dt =d[A] dt →r=−1 a d[A] dt =... 1 e d[E] dt =... (1.5) where [i]indicates the molar concentration of the species i=A,B,E.... For most systems, it was experimentally demonstrated that the reaction rate at a given time is related to the concentration of the involved species: r=k[A]α[B]β···[L]λ(1.6) where kis known as the reaction rate (or kinetic) constant and, the exponents α,β,...,λare the partial orders of the reaction. α+β+···+λ≡nis known as the overall order of the reaction. The expression of the reaction rate rshown in Eq. (1.6) is known as the law of mass action [116]. 2 DAR´ IO MART´ IN ESCALA VODOPIVEC 1.2 Types of Reactions The present section will briefly introduce several types of reactions. All of them are important to the development of this thesis. Elementary Reactions: An elementary chemical reaction consists of a single step, in which no intermediate compounds are observed. The chemical transformations occur in only one single step and pass through a single transition state [116, 13]. In these reactions, the molecularity and the reaction orders are well defined and can be derived from the stoichiometry. There are several types of elementary reactions, the most common ones are the unimolecular (first-order) and bimolecular (second-order) reactions. The dynamics of these reactions are modeled by the mass action law, that assumes that the rate of the elementary reaction is directly proportional to the product of the activities or concentrations of the reactants [116]. In a first-order elementary reaction, a molecule or a chemical species dissociates, polymerizes, or directly converts into one or more products (P). These reactions can be generally expressed as: AkP (R1) where the reaction rate is given by: r=−d[A] dt =k[A](1.7) Some examples of unimolecular reactions are: O3O2+O (R2) H2O2(l)H2O(l)+0.5O2(g)(R3) On the other hand, in second-order elementary reaction, two species A and B react to produce one or more products (P). These reactions can be expressed generically as follow: A+BkP (R4) whose reaction rate is given by: r=−d[A] dt =−d[B] dt =k[A][B](1.8) A simple example of bimolecular reaction can be: C12H22O11 +H2OkC6H12O6+C6H12O6(R5) Complex chemical reactions: a complex chemical (or multistep) reaction is a chemical process that consists of several steps of elementary reactions and has one or more reaction intermediaries. These processes must be described by reaction mechanisms detailing all the involved elementary steps. The rate law of a complex chemical reaction is obtained by combining the rate laws of the multiple elementary steps, where every elementary step follows 3 Chapter 1. Introduction was developed at the University of Oregon by Field and Noyes in 1974 [67]. The model was obtained as a direct reduction of the FKN mechanism and it was achieved by the application of standard methods of chemical kinetics like the rate-determining-step approximations [201, 183, 130]. In contrast to the Brusselator, the Oregonator model includes three dynamic variables (the inhibitor, the activator, and the catalyst), and an adjustable kinetic modulator. This makes the model much richer in terms of the complexity and behaviors that can be reproduced by it, like for example, bistability and chaos in CSTR [81, 80, 164]. The Oregonator is described by the following set of chemical equations: A+Yk1X+P (R18) X+Yk22P (R19) A+Xk32X (R20) 2X k4A+P (R21) B+Zk51 2fY (R22) where kinetic equations are given by: d[X] dt =k1[A][Y]+k2[X][Y]−k3[X][A]−2k4[X]2 d[Y] dt =k1[A][Y]−k2[X][Y]+ 1 2k5f[B][Z] d[Z] dt =2k3[A][X]−k5[B][Z] (1.19) where X represents the activator, Y the inhibitor, and Z the catalyst which corresponds to HBrO2, Br–and Fe2+ from the original BZ reaction respectively. fis an adjustable stoichiometric factor that makes it possible to modulate the dynamics of the reactions. Example simulations of both, the Brusselator and the Oregonator kinetic models are presented in Figure 1.5. These simulations were done by using the GNU software COPASI [95]. 1.4.2 The 1,4-Cyclohexanedione-Bromate-Acid Oscillatory Reaction As it was mentioned, one of the secondary products of the original BZ formulation is CO2 (see reaction R113). The formation of carbon dioxide can be appreciated as tiny bubbles that emerge from the BZ solution. In some cases, this can be detrimental for some studies, as the bubbles may, for example, interact with the measurement equipment or hinder the subsequent analysis of the experiments. In order to overcome such an issue, in 1994, Kurin-Cs¨ orgei et al developed an alternative formulation of the classical BZ reaction by replacing the original organic substrate with 1,4-Cyclohexanedione (hereafter CHD - Fig. 1.6) [114, 187, 188, 190, 189]. This alternative formulation is usually called the BZ-CHD reaction of Bubble-Free BZ reaction. The replacement of the malonic acid not only eliminates the formation of carbon dioxide 10 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 1.5: Simulations of the temporal dynamics of (a) the Brusselator and (b) the Oregonator kinetics models. In both cases [X] and [Y] represent the molar concentration of the activator and the inhibitor species respectively. In (b), [Z] represents the molar concentration of the catalyst species but also produces dynamics significantly different compared to those observed in the original BZ system [123, 84, 122]. Thus, traveling shock structures, long-lived oscillations, light sensitivity, wave merging, stacked wavefronts, densely packed patterns, and segregated clusters are examples of the broad variety of behaviors observed in the bromate-CHD-ferroin system [190, 189]. All these characteristics made the BZ-CHD to be considered superior over the classical BZ reaction, and more suitable to study multitude of different phenomena [190, 188, 187, 189]. O OH OH O (a) Malonic Acid (MA) O O (b) 1,4-Cyclohexanedione (CHD) Figure 1.6: Organic substrates for (a) the original BZ reaction, (b) the BZ-CHD reaction Several studies were made to understand and fully characterize the mechanism of the BZ-CHD reaction [113, 112, 187, 188, 114, 190, 189]. An extensive kinetic model was also proposed by Szalai et al, where the dynamics and properties of the BZ-CHD reaction were studied for different catalysts and for the uncatalyzed reaction [189, 190]. Similar to the FKN model, reduced or skeleton models were also developed. In such models, the main characteristics of the systems are preserved even with a reduced number of variables. This makes possible its numerical implementation without requiring sophisticated computational resources [189, 190, 188]. The kinetic models of the BZ-CHD reactions are not indicated in this introductory section, as they will be extensively used and described in the forthcoming chapters. For this thesis, the ferroin catalyzed BZ-CHD reaction will be preferred over the classical BZ formulation. This reaction will be extensively used in Part I to study its coupling with hydrodynamic systems. 11 Chapter 1. Introduction 1.5 pH-Shifting Reactions A pH-shifting reaction is defined as a system in which the hydrogen ion plays the most important kinetic role in the overall dynamics. In these systems, the driving force of such changes is the variation in pH, which can be as large as 6 pH-units [145, 153]. pH-shifting reactions are a well-known group of reactions that exhibit relatively simple chemistry. The reaction mechanism in these systems is better understood compared to other types of complex reactions [145, 153]. This is possible since the stoichiometry and the kinetics of the global and intermediary reactions were thoroughly studied, making it possible to easily identify the positive and negative feedback processes. As was mentioned in Section 1.3, some complex reactions change their dynamics depending on the experimental conditions. This is the case for most pH-shifting reactions[145]. Some formulations exhibit oscillations or complex dynamics when studied in CSTR or semibatch reactors, and only a single pH switch when the same reaction occurs in a batch reactor. When producing oscillations, the pH-shifting reactions are also known as pH-oscillators [145, 153, 129]. There are two main categories of pH-shifting reactions that will be addressed in this introduction. One are the so-called inorganic, in which the key components are inorganic, and the other are the so-called organic (or special) in which part of its formulations are composed of organic species. 1.5.1 Inorganic pH-Oscillators Inorganic pH-oscillators were well known and vastly studied. In most cases, studies were focused on the oscillating dynamics observed in CSTR or semibatch reactors. These reactions are usually composed of an oxidant and a reductant species, thus they are redox processes. A broad description of all known pH-oscillators is presented in Orb´ an et al [145]. Additionally to the oxidant and the reductant, in some systems, a second substrate is necessary for the oscillations to occur [145]. The dynamics of a pH-oscillator will depend on whether a second substrate is needed or not [145, 153]. Both situations are schematized in Figure 1.7. In the one-substrate systems, the reductant is oxidized by a partial oxidation mechanism producing a reaction intermediary. This step is proton-consuming (negative feedback), thus it produces an increment in the pH (Step 1, Fig. 1.7(a). In a second step, the intermediary will react with both, oxidant and reductant through a total oxidation mechanism. In this step apart from the end product, protons are generated by an autocatalytic mechanism decreasing the pH (Positive feedback, Step 2, Fig. 1.7(a)). Oscillations are produced by the repetition of steps 1-2. The cycle will continue until the substrates are consumed. In a two-substrate system, oscillations occur as a consequence of two consecutive reactions. In the first step, protons are produced autocatalitically by an oxidation mechanism causing the pH to decrease (positive feedback, Step 1, Fig. 1.7(b)). The generated protons are consumed in a second step in which another substrate is oxidized to produce the end product (negative feedback, Step 2, Fig 1.7(b)). As this is a proton consuming reaction, the pH increases and the cycle starts again. Oscillations will continue until the first substrate is consumed. The second substrate can be a reductant or a H+consuming reagent. An experimental example of the dynamics of a one-substrate oscillator is presented in 12 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 1.7(c)). These results were obtained from the BrO3–/SO3– 2 system [145]. In this particular case, a small amount of Mn+2 was added as a catalyst. The reaction was performed in a semibatch reaction at 45 °C [153]. The changes in the pH were followed by using a pH-meter. Inorganic pH-oscillators can be modeled by considering a simple skeleton mechanism. This model was suggested by R´ abai et al [158, 129] and it consists on three steps: A−+H+HA (R23) B+HA H+ H++P (R24) C+H+Q (R25) where A–is the conjugate base of a weak acid (HA), the reductant. This species is oxidized by B to a conjugate base of a strong acid (P). The second substrate is indicated by C and the end product is Q. The positive feedback is represented by the autocatalytic step in the second reaction. The third reaction is the negative feedback which is the proton consuming step. This simple model can reproduce the oscillatory dynamics of the pH oscillator both in homogeneous and spatially extended systems [129]. pH-Oscillators are attractive, not only for their simplicity but also for their potential applications. One of these applications is their use for drug delivery in living systems. The fundamentals of such techniques will be not addressed here, but the idea is to couple a pH-sensitive material to the dynamics of the pH-shifting reaction to create a chemo-mechanical device. In this way, a specific drug or compound could be specifically released due to changes in the conformation of the material produced by dynamics in the pH [36, 174, 14, 118, 203, 5, 196, 110]. Even though the idea is promising, there is a major problem: Most of the pH-sensitive materials are often organic, and these inorganic pH reactions result too aggressive for such types of molecules [107]. To overcome this issue, organic pH-shifting reactions were created Figure 1.7: Schematics of the mechanism of the (a) one-substrate and (b) two-substrate pH-oscillators. (c) Experimental results of one-substrate pH-oscillator. This experiment was done in a semibatch reaction at 45 °C following the recipe indicated in Poros et al for the BrO3–/SO3–2-Mn+2 system [153]. All pH-changes were recorded by using a pH-meter. 13 Chapter 1. Introduction and will be discussed in the next section. 1.5.2 Organic pH-Shifting Reactions Organic pH-shifting reactions are a special type of pH reactions in where the changes in the pH are not produced by redox processes but by acid-base steps [145]. There are two main reactions that will be used in this thesis: the Formaldehyde-Sulfite (FS) and the Formaldehyde-Sulfite-Gluconolactone (FSG) reactions. Both reactions will be introduced from a completely descriptive point of view. As these reactions will be extensively used in Part II of the thesis, more information related to their modeling and characterization will be addressed in their corresponding chapters. The Formaldehyde-Sulfite Reaction The FS reaction is a well-known pH-shifting reaction where the pH rapidly changes from acid to basic after an induction period [212, 213]. This phenomenon received the name of clock behavior due to the abrupt change in the pH. The reaction consists of a buffer composed of sulfite (SO32– ) and bisulfite (HSO3–), and formaldehyde (in the form of methylenglycol, which is hydrated form of formaldehyde). The clock mechanism is produced by the consumption of the internal sulfite/bisulfite buffer by formaldehyde to produce hydroxymethanesulfonate, which is a formaldehyde-sulfite adduct. The maximum pH transition is about 4 units of pH (from 6 to approximately 10). Both, the induction time and the pH jump can be modulated by changing the initial concentrations of the chemical species and the sulfite/bisulfite proportion. The typical clock behavior is observed in a batch reactor, however, oscillations and complex dynamics were observed in CSTR [106]. An example of the experimental FS clock reaction is presented in Figure 1.9(a). As can be seen, the pH changes after the induction period. Both, the initial and the final pH depend on the initial species concentrations. This reaction is extensively studied in Kovacs et al [106]. In such work, the system is completely characterized and modeled, showing the changes in the dynamics produced by the type of reactor where the reaction occurs. The Gluconolactone-Formaldehyde-Sulfite Reaction The FSG reaction was created from the FS reaction to induce delayed negative feedback in the dynamics of the reaction [108, 107]. The reaction shares the same composition as the FS reaction, but including one more species, the D-(+)-gluconic acid δ-lactone (or gluconolactone). The characteristic clock dynamic of the FS reaction is coupled with the hydrolysis of the gluconolactone that produces gluconic acid. This process is schematized in Figure 1.8. The hydrolysis reaction is base-catalyzed, which means that the OH–generated after the consumption of the sulfite/bisulfite buffer in the FS reaction, will increase the rate of hydrolysis. The coupling between these two processes produces a single peak in the pH, which indicates both positive and negative feedback. An experimental example is presented in Figure 1.9(b). In CSTR this reaction also exhibits complex behaviors, such as oscillations [108, 107]. Figure 1.10 shows an experimental example of the FSG reaction conducted in a continuous stirred tank reactor. As can be appreciated, the oscillations are indicated by the change in the color indicator (Bromothymol blue). In the acid state, this indicator shows a yellow coloration 14 DAR´ IO MART´ IN ESCALA VODOPIVEC O OH OH OH O OH OH– H2OHO O OH OH OH OH OH Figure 1.8: Schematics of the hydrolysis of the gluconolactone (left) to gluconic acid (right). This process is base catalyzed, which means that the OH–produced after the buffer consumption in the FS reaction increases the rate of hydrolysis. Figure 1.9: Examples of the dynamics of (a) FS reaction and (b) FSG reaction in batch reactor. The FS reaction is characterized by the abrupt change in the pH after the induction time produced by the consumption of the SO32– /HSO3–buffer. In the FSG reaction, the dynamics observed in (a) are coupled with the hydrolysis of the gluconolactone, providing the delayed negative feedback necessary to produce the single peak observed in a batch reactor. that turns into blue in the basic state. The oscillations were recorded with a pH-meter connected to a computer. As can be seen, the oscillations are formed by the cycling process produced by the buffer consumption and the hydrolysis of the gluconolactone. As the substrates/products are continuously added/removed to the system, the single peak dynamic became oscillating. Figure 1.10: Experimental example of the FSG reaction in a CSTR. (a) Oscillations are easily observed by following the changes in the color indicator, or (b) by using a pH-meter. 15 Chapter 1. Introduction 1.6 Reaction-Diffusion-Convection (RDC) Systems The previous sections described different types of reactions in stirred systems. However, it is not always possible to homogeneously distribute a mixture. In such cases, it is important to consider the effects produced by diffusive and convective forces in the active medium. Diffusion manifests in systems when specific properties are not homogeneously distributed in the medium. For example, the mixing of a droplet of ink in a glass of water is produced by the random collisions of the ink molecules with the water molecules. The effect of diffusion in an inhomogeneous system is responsible for many complex behaviors that are part of the study of this thesis. On the other hand, many processes in Nature and the industry occurs under the effect of advective forces, like the transport of pollutants in fluid flows [146], convective plumes in the ocean [200, 111], dispersion of aerosol in the atmosphere, mixing processes in reactors [131], etc. More specifically, the transport of a chemical species in a reactive medium combined with diffusivity can be responsible for a multitude of different complex phenomena [179]. A mathematical description for the transport of a chemical species can be derived from the general continuity equation as [179]: ∂Ci ∂t+∇·~ Jtot =Ri(1.20) where ~r= (x,y,z)is the position vector, Ci=Ci(~r,t)are the concentrations of the species i in a specific position and time, Riare the net volumetric source of each the species i, and ~ Jtot =~ Jdi f f +~ Jadv is the total flux produced by the diffusive (~ Jdi f f ) and the advective (~ Jconv) flux. The expression 1.20 relates the rates of change of every species ito the flow and diffusion into and out of a differential control volume and considers the generation or consumption inside the control volume [179]. The diffusive flux is produced by the diffusion of the molecules moving randomly through the medium, and can be described by Fick’s first law: ~ Jdi f f =−Di∇Ci(1.21) where Di=Di(~r,t)are the diffusion coefficient of each species i. On the other hand, the advective flux is associated with the flow convection by the following expression: ~ Jadv =~uCi(1.22) where ~u=~u(~r,t)is the velocity field. The combination of Eqs. (1.21)-(1.22) with Eq.(1.20), gives the general transport equation for each chemical species: ∂Ci ∂t+∇·(−Di∇Ci+~uCi) = Ri(1.23) Considering the reactivity between the chemical species, the source and sink terms Riare now dependent on the concentration of the species, therefore Ri=fi(Ci,t), where fiare the net 16 DAR´ IO MART´ IN ESCALA VODOPIVEC reaction rates of the chemical system. By combining Equation (1.1), and considering that the diffusion coefficients do not depend on the position and time, Equation (1.23) is reduced to: ∂Ci ∂t+~u·∇Ci=Di∇2Ci+fi(Ci,t)(1.24) which is known as the reaction-diffusion-convection equation for the species i. In a non-reactive system, the source and sink term can be neglected obtaining: ∂Ci ∂t+~u·∇Ci=Di∇2Ci(1.25) This equation is known as the diffusion-convection equation and it will be fundamental to explain non-reactive convective processes [179]. In this thesis, Eqs. (1.24) and (1.25) will be fundamental in the development of numerical models. 1.6.1 Reaction-Diffusion (RD) Systems Reaction-diffusion systems are mathematical models that describe how spatially extended species are affected by two main processes. One is the diffusion, that facilitates the spatial distribution through the medium. The other one is the chemical process that modulate the creation and/or consumption between species. Many natural phenomena can be modeled by reaction-diffusion systems, such as somitogenesis[149], animal pigmentation[147, 105], cell differentiation [38], and chemotaxis [180] among others. More specifically, non-linear reaction-diffusion systems are in general, useful to model out-of-equilibrium phenomena like self-organization and spatio-temporal pattern formation of biological systems. These problems are mainly driven by spontaneous symmetry breaking in inhomogeneous media [156, 155, 20, 115]. There are three main characteristic regimes associated with RD systems: excitable [40, 152, 133], bistable [40], and oscillating [55, 54]: Excitable regime: In an excitable medium, a local perturbation of the species propagates through the medium. This situation is typically observed in cardiac and nervous tissues, neurons, and in some diseases like Parkinson’s or epilepsy. In an inhomogeneous chemical system, the excitable regime is manifested in the form of circular, planar, or triangular traveling waves or in the form of a spiral [152, 133, 60], which are often produced by the rupture of a concentric wave. Excitable regimes are also characterized to show, in some specific conditions, structures that are stationary in space. An example of this is presented in Figure 1.11(a). These structures are known as Turing patterns and were obtained from the BZ-AOT chemical system3[205]. Turing patterns were used as the standard model to explain the pigmentation in animals [205]. These types of structures will be not considered in this thesis. Oscillating regime: In an oscillating regime, the local perturbation propagates through the medium similar to the excitable case, but periodically. There are plenty of oscillating systems in Nature. Some examples can be found in population dynamics, metabolic cycles, cardiac and 3Author’s Note: these Turing patterns were the first to be obtained experimentally at the Universit´ e Libre de Bruxelles, home of the Brusselator model. They were obtained by the author during his pre-doctoral stay in 2013-2014. 17 Chapter 1. Introduction circadian rhythms, and many more. In particular, the oscillating chemical reactions introduced in Section 1.4 are examples of non-linear chemical oscillating systems. Figure 1.11(b) shows an example of reaction-diffusion patterns observed in an inhomogenous BZ-CHD reaction. Bistable regime: The bistable regime is characterized by the coexistence of two different states configuring a stable spatial concentration distribution. The importance of these systems lies in their capacity to store information since they have two stable states. This was observed in the mechanism used by neurons when discriminating nerve impulses [167]. Bistability is not common in chemical systems, however, there are some examples of reactions that exhibit such behavior [19, 97]. Figure 1.11: Examples of spatio-temporal structures obtained from reaction-diffusion systems. (a) Formation of spatially stationary Turing patterns observed from the BZ-AOT system. (b) Traveling waves and spirals observed from the BZ-CHD reaction. A general mathematical description for reaction-diffusion systems can be derived from Eq. (1.24) by neglecting the terms associated with the convection: ∂Ci ∂t=Di∇2Ci+fi(Ci,t)(1.26) this expression is known as the reaction-diffusion (RD) equation for the species i. Many systems in Nature can be modeled in a simplified manner by considering only two variables, an activator (C1) and an inhibitor (C2). This is possible by the principle of slaving [83], which eliminates all variables that quickly converge to the stationary state and therefore, have little effect on the dynamics of the system. Thus, a more specific mathematical model based on this principle is described by: ∂C1 ∂t=D1∇2C1+f1(C1,C2,t) ∂C2 ∂t=D2∇2C2+f2(C1,C2,t) (1.27) where the non-linear behavior of the system is determined by the expression of the reaction terms f1and f2. The existing numerical models for the BZ reaction introduced in Section 1.4.1, are capable to reproduce the dynamics of the excitable and oscillatory regimes. This can be achieved by combining Eqs. (1.27) with the mathematical models described by Eqs. (1.18) and (1.19). 18 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 1.12 shows the results of simulating the Brusselator and Oregonator models with diffusion. Spatio-temporal patterns similar to those observed in Figure 1.11(b) are presented in Figure 1.12(a). These structures were obtained by simulating the spatial Brusselator model. In this case, the system is under the oscillating regime, and it is characterized to show concentric traveling waves with global oscillations at the center. Figure 1.12(b) shows an example of the excitable spiral obtained by simulating the spatial Oregonator model. In this case, there are no oscillations and only one single structure propagates through the medium. Figure 1.12: Simulations of (a) oscillating regime obtained by numerical simulation of the spatial Brusselator model, and (b) excitable spiral obtained by numerical simulation of the spatial Oregonator model. Reaction-diffusion patterns can be also obtained from pH-oscillators. These reactions are especially suitable to produce stationary structures due to their autocatalytic nature. In these systems, the role of the activator is played by the H+ion, and the inhibitor will depend on the specific formulation [145]. Similar to the BZ reaction, patterns can be observed depending on the relative diffusivities of the inhibitor and the activator. When both are approximately equal, traveling waves are favored. Turing-like spatio-temporal structures can be obtained by decreasing the diffusivity of the activator. This can be achieved by introducing large anions like the polyacrylate (Section 1.9), which produces a reversible binding effect of such a molecule [129]. Another interesting characteristic is that any obtained structure can be observed with acid-base indicators without any other sophisticated method [186, 145]. There are more complex situations on the bibliography where reaction-diffusion systems are controlled or modified by external forcings like centrifugal forces [79, 60], advection [209, 208], electric fields [151], and temperature[30]. 19 Chapter 1. Introduction 1 aZa 0 µ∇2~vdz =1 aZa 0 (µ∇2~u f (z)−~u f 00(z))dz =µ∇2~u−12µ a2~u(1.41) are obtained the so-called Brinkman equations: ∇p≈ −12µ a2~u+µ∇2~u+ρ~g(1.42) Brinkman equations are valid for low velocity flows in porous and non-porous media. Due to the intrinsic characteristics of the Hele-Shaw flows, it is possible to assume that the friction term dominates over the viscous term [34, 33, 93, 45]. Taking this consideration, Eq. (1.42) can be reduced to Darcy’s Law: ∇p≈ −12µ a2~u+ρ~g=−µ κ~u+ρ~g(1.43) where κ=a2/12 is the intrinsic permeability of the Hele-Shaw cell. From Equation (1.43) it is possible to observe that the porosity of a Hele-Shaw cell is a constant φ= 1. Additionally, from the simplifications and assumptions made on the original Navier-Stokes equation, it was also possible to obtain an expression for the intrinsic permeability of the cell. As can be deduced from Equation (1.43), the flow inside a Hele-Shaw cell is representative of the flow in a porous medium. Thus, the study of such flows can be done using such devices in a controlled environment. Moreover, an advantage of using Hele-Shaw cells is that, since they are transparent, they allow a perfect visualization of fluids inside the cell, which is usually a major limitation in porous media. 1.8 Fingering Instabilities in Hele-Shaw Cells Having introduced the hydrodynamic instabilities and once explained the main aspects of the flow in porous media, Darcy’s law, and the flow inside a Hele-Shaw cell, the next sections will introduce the fingering phenomenon in Hele-Shaw cells. This thesis aims to analyze the coupling of these specific groups of instabilities and complex chemical reactions. However, it is necessary to briefly introduce first the main aspects of the non-reactive classical, and the newer chemically-driven instabilities, as a way to review the state of the art at the beginning of this work. This general introduction will address descriptively the main general aspects related to the fingering instabilities, including experimental cases and numerical simulations as reference examples. 1.8.1 Density Fingering Instability Density fingering (or Rayleigh-Taylor) instabilities are observed when fluids of different densities are subjected to acceleration in a direction opposite to that of the density gradient [33]. This situation is often observed when a denser solution lies on top of a lighter one in the gravity field (similar to the example presented in Figure 1.13(d)). [171, 200, 111]. A general scheme of the Rayleigh-Taylor instability for two solutes A and B is presented in Figure 1.18(a). 26 DAR´ IO MART´ IN ESCALA VODOPIVEC If the fluids are miscible, this instability is triggered when the density of the solution is affected by the spatial changes in the concentration of the solute dissolved (A=A(x,y,t)and B=B(x,y,t)). In this situation, the density of the system is governed by the concentration of such solutes (ρ=ρ(A,B)). The main stability factor of the system is the density difference between the two solutions (∆ρ=ρA−ρB). The stability of the system increases proportionally with the density jump between the upper and lower layer, if such a jump is favorable. This is, denser fluid below a lighter one. The opposite situation produces the destabilization of the fluid interface. A mathematical model for the density fingering instability can be obtained by combining the equation of the flow inside the Hele-Shaw cell (Eq. (1.43)) with the reaction-diffusion-convection equation for solutes A and B (Eq. (1.25)). Assuming that the flow is governed by Darcy’s law and no 3D effects occur along the cell gap, the general equations of the problems are given by: ∇·~u=0 ∇p=−µ κ~u+ρ(A,B)~g ∂A ∂t+~u·∇A=DA∇2A ∂B ∂t+~u·∇B=DB∇2B (1.44) The validity of this equation is founded on the Boussinesq approximation [22], which assumes that all the density variations induced by concentrations (and/or temperature) changes are small compared to the mean density. As a consequence, the condition of incompressibility (∇·~u=0) of the fluid is preserved, as the buoyancy effects only retain in the ρ~gterm of the flow equation. Therefore, any local pressure variation produced by small density changes can be neglected. The mathematical model presented in Eqs. 1.44 can be solved numerically to reproduce the physical phenomenon. Figure 1.18(b), shows the non-linear simulation of a typical Rayleigh-Taylor instability obtained by CFD software. As can be seen, compared to the experimental situation shown in Figure 1.18(c), the simulations agree very well in both the system dynamics and the shape of the fingers, validating at least qualitatively, the numerical model [197, 121, 45]. Until now, it was only considered the unstable case, however non-reactive buoyancy-driven instabilities can be obtained even for initially stable configurations. In this case, the diffusivity also plays an important role in the destabilization of the system. Figure 1.19(a) schematizes a situation in where the denser fluid is on the bottom of the Hele-Shaw cell, and the lighter one is on top. This situation is initially stable, but fingering may be observed if differential diffusion exists, which means that one of the solutes diffuses faster compared to the other one. In the case where the denser fluid diffuses faster than the lighter one, double-diffusive fingering instability is obtained (DD, [197]). An example of this instability is presented in Figures 1.13(d-d2). It can be found in Nature when hot saline water lies over cold freshwater of a higher density. This instability is responsible to improve the transport of nutrients and/or to control the temperature in the oceans [171, 200, 111]. This case can be modeled using the same set of equations used for the Rayleigh-Taylor instability, but considering proper values for DAand DB. Some numerical results of this case 27 Chapter 1. Introduction Figure 1.18: Rayleigh-Taylor Instability (a) Schematics of the initial fluid and solute configuration. In this instability, the denser fluid is on top of the lighter one, leading to an unstable hydrodynamic initial condition. (b) Image sequences of a Non-linear simulation of the Rayleigh-Taylor instability obtained by numerical integration of Eq. 1.44. (c) Image sequences of a Rayleigh-Taylor instability obtained experimentally. In this particular case, a denser solution of ferroin is put on top of doubly distilled water. are presented in Figure 1.19(c). The driving force of this instability is the local destabilization induced by the faster diffusive species [197, 51, 45]. Figure 1.19: Density Fingering Instability. (a) Schematics of the initial condition. (b) Non-linear numerical simulations of the diffusive-layer convection (DLC) instability. (c) Non-linear numerical simulations of the double-diffusive (DD) instability. All simulations were performed by numerically integrating Eqs. 1.44 on CFD software. On the other hand, if the lighter fluid diffuses faster compared to the denser one, diffusive-layer convection (or DLC, [197]) instability is obtained. A numerical example of this situation is presented in Figure 1.19(b). Wooding et al [217] demonstrated that the effect of the diffusion in miscible fluids is analogous to the role played by the surface tension in the immiscible case, which is to set 28 DAR´ IO MART´ IN ESCALA VODOPIVEC the initial length scale of the fingering pattern. There are many works where the classical buoyancy-driven instabilities were completely analyzed and characterized by theoretical and/or experimental perspectives [59, 87, 8, 9, 120, 44, 25, 197]. 1.8.2 Viscous Fingering Instability Viscous fingering instabilities are observed when a less viscous fluid displaces a more viscous one in a porous medium. This phenomenon is also known as Saffman-Taylor instability when fluids are immiscible [93]. When fluids are miscible, the viscous fingering instability is driven by viscosity gradients which results from spatial variations in the concentration of the solutes that govern the viscosity of the solution such as µ=µ(A,B). This instability was experimentally and theoretically studied, for both, linear [168, 46, 41], and radial [93, 168, 88, 141, 44] displacements. In viscous fingering, the stabilization of the system is governed by the log mobility ratio R, which is defined as [48, 193]: R=lnµA µB(1.45) The instability occurs when R>0 [44, 45]. The stability of the system decreases proportionally to the increment in R. Similar to the density fingering, the mathematical description of the viscous fingering instability is based on Darcy’s law but considering the viscosity changes produced by concentrations of the involved solutes. This statement coupled with the transport equation for solutes A and B (Eq. (1.25)), allows to obtain the governing equations for the non-reactive viscous fingering instability in a Hele-Shaw cell: ∇·~u=0 ∇p=−µ(A,B) κ~u ∂A ∂t+~u·∇C=DA∇2A ∂B ∂t+~u·∇C=DB∇2B (1.46) In this case, as the displacement occurs horizontally and the cell gap is narrow, the buoyancy forces can be neglected [93, 45]. Thus, the corresponding term is included in the pressure gradient. Examples of experimental and numerical radial viscous fingering are presented in Figure 1.20. In Figure 1.20(a), a more viscous and colored solution of Polyethylene glycol (PEG-300) is displaced by doubly distilled water. As can be appreciated, the displacement is unstable and viscous fingering occurs. Figure 1.20(b) presents the non-linear simulations of a viscous fingering instability obtained by integrating numerically Eqs.(1.46) by a CFD software. As can be seen, the shape of the numerical fingers is very similar to the experimental counterpart, validating one more time the mathematical model derived from Darcy’s law. 29 Chapter 1. Introduction Figure 1.20: Non-reactive viscous fingering instability in a radial Hele-Shaw cell. (a) An experimental case where a Polyethylene Glycol (PEG-300) colored solution is displaced by doubly distilled water. (b) Non-linear simulation obtained by integrating Eqs. (1.46) and performed with CFD software suite. A Schematic of the reagent and viscosities configuration is presented in the second snapshot of (b). Diffusion can also trigger a viscous fingering instability in an initially stable configuration. Mishra et al ([128]), studied these diffusion-driven viscous instabilities for a broad variety of parameters. Figure 1.21 shows the numerical results of three different situations. In all cases, it is assumed that one of the species diffuses faster than the other one (i.e. DA>DB), and the initial fluid configuration is stable (i.e. µA>µB). In Figure 1.21(a), patterns are obtained due to a differential diffusion mechanism induced by the destabilizing effect the slower species. This situation is known as DNS-VF [128]. When the instability is produced by purely double diffusive mechanism similar to the buoyancy-driven case (Fig. 1.19(c)), the instability is known as DD-VF. This situation is shown in Figure 1.21(b). Figure 1.21(c) shows the situation in which patterns are obtained due to a differential diffusion mechanism induced by the destabilizing effect the faster species. This case is known as DNF-VF [128]. These three cases are obtained by varying some numerical parameters associated with the concentrations and viscosities of the displacing and displaced fluid[128]. The figure is included as an example of the versatility and the broad possibilities related to the non-reactive viscous fingering and its modeling. 30 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 1.21: Numerical results of a diffusion-driven non-reactive viscous fingering instability for obtained for linear displacement. (a) Viscous fingering originated by the destabilizing effect of the slower species and induced by differential diffusion (DNS-VF) (b) Viscous fingering originated by pure diffusive effects, similar to the double diffusion instability (DD-VF) (c) Viscous fingering originated by the destabilizing effect of the faster species and induced by differential diffusion, similar to the diffusive-layer convection instability (DNF-VF). In all cases DA>DBand µA>µB[128]. 1.8.3 Chemically Driven Fingering Instabilities The study of chemo-hydrodynamic fingering instabilities added a new step in complexity by coupling chemical processes to the flow in porous media. The dynamics of the fingering phenomenon in reactive systems mainly depends on the interaction between the species involved and the flow. If the species are passively advected, the properties of the fingering interface remain those of the non-reactive case presented in the previous sections. This occurs even considering that the flow clearly influences the spatial-temporal distribution of chemical instability. However, there are cases where the chemical reaction plays an active role in the development of the fingering instability, especially when they can actively influence or trigger convective motions when two solutions containing separate reagents come into contact. The key to controlling chemically driven instabilities lies in the effect on the viscosity or density profiles produced by the localized generation of products due to interfacial reaction. These changes affect the mobility of the fluids by favoring or reducing instability. 31 Chapter 1. Introduction The following introduction will be consider simple A+B→Creactions as the starting point for the development of more complex situations, which is the main objective of this thesis. Thus, the next sections will present a brief outline of the most common chemically induced fingering instabilities, including experimental and numerical examples. Density Fingering In addition to all the physics associated with the non-reactive case, the inclusion of a new species (C) with different density and diffusivity, can strongly alter the dynamics of the system in buoyancy-driven instabilities. The variation in the density produced by the local generation of the reaction of the product can force changes in the convection field or even mixing. This situation is illustrated in Figure 1.22(a) for an initially stable case. A mathematical model for this type of systems can be derived from the classical non-reactive case (Eqs. (1.44)), by including the corresponding reaction terms in the transport equations, and the expression for the density changes produced by reactants and/or products. ∇·~u=0 ∇p=−µ κ~u+ρ(A,B,C)~g ∂A ∂t+~u·∇C=DA∇2A−kAB ∂B ∂t+~u·∇C=DB∇2B−kAB ∂C ∂t+~u·∇C=DC∇2C+kAB (1.47) Figure 1.22(b,c) shows numerical examples of chemically driven density fingering in two hypothetical situations. Figure 1.22(b), presents the case where C is lighter than Aand B, thus it floats to the upper part of the reactor/domain. This situation can be observed experimentally in instabilities driven by neutralization reactions [65, 9]. The second case is presented in Figure 1.22(c), where C is denser than Aand B. In this situation, the product of the reaction sinks to the bottom part of the reactor/domain. This can be observed experimentally if the instability is driven by a precipitation reaction [44, 23, 138, 176]. The model for ρ=ρ(A,B,C)will depend on the nature of the system and the chemical species. Thus, in some situations, it is useful to consider all the species involved [45], in other situations like the one exemplified in Figure 1.22, only the effect of C. In any case, the expression of ρis based on the Boussinesq approximation assuming a linear relationship between the concentration of the species and the density [197, 45]. Viscous Fingering In the case of chemically induced viscous fingering, the interplay between reaction and the instability occurs through changes produced on the viscosity. Thus, a chemical reaction can produce an increment or decrement on the viscosity at the miscible interface, affecting the properties of the fingering pattern. The development of a model for reactive viscous fingering is analogous to the density fingering case. The only difference is the necessity to combine the flow and transport equation with a suitable model for the viscosity changes due to the reactive species. Similar to the density 32 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 1.22: Numerical results of a density fingering instability induced by the reaction A+B→C. In this case, the density is assumed to be affected primarily by the product C. (a) Schematics of the fluid configuration and the spatial location of the chemical species. The system is initially stable and it is destabilized by the product of the reaction between Aand B. (b) The product Cis lighter compared to the remaining fluids present in the reactor. This produces the fingers to ascend to the upper part of the reactor. (c) Same situation as (b), but Cis denser. In this case, the fingers sink to the bottom of the reactor. All simulations were performed by using CFD software. case, the expression of the viscosity function depends on the nature of the problem. Considering these statements, a general description for this problem is given by: ∇·~u=0 ∇p=−µ(A,B,C) κ~u ∂A ∂t+~u·∇C=DA∇2A−kAB ∂B ∂t+~u·∇C=DB∇2B−kAB ∂C ∂t+~u·∇C=DC∇2C+kAB (1.48) Figure 1.23 shows two examples of viscous fingering affected by a chemical reaction for an experimental case (Fig. 1.23(a)) and a numerical case (Figure 1.23(b)). Even though this instability is not produced by the reaction itself, as the hydrodynamic scenario is unstable. However, this is a simple example of how a chemical reaction can interact with a fluid displacement. Figure 1.23(a) shows a more viscous polymeric solution of Poly(acrylic acid) and sodium bisulfite is displaced by a less viscous aqueous solution of formaldehyde. The polymeric solution is colored with a pH color indicator. The reaction locally increases the pH of the system at the interface producing the color indicator to change from yellow to blue. The system remained unstable, but fingers started to exhibit a blue coloration that propagates due to the effect of the diffusion. This situation was simulated by modeling the chemical reaction with the simple A+B→C. Results are presented in Figure 1.23(b). As it is possible to see, both numerical and experimental dynamics present much similarities. 33 Chapter 1. Introduction Figure 1.23: Numerical results of a viscous fingering instability affected by the reaction A + B →C. In this case, the system is initially unstable. The reactivity is evidenced by the change in the color indicator at the interface between fluids A and B. (a) Experimental case where a colored polymeric solution of Poly(acrylic acid) and sulfite is displaced by formaldehyde. The contact between these two solutions produces an increment in the pH that changes the yellowish coloration of the indicator into blue. (b) Simulation of a viscous fingering instability coupled with an A + B →C reaction. Even though the shape of the fingers is not exactly equal to the one observed in the experimental case, the physical phenomenon associated with the reaction is well reproduced. Chemical reactions can also induce viscous fingering instabilities by changing the permeability of the porous matrix. This could be produced by precipitation [86, 176] and dissolution [102] reactions, for instance. In this case, once the displacing and displaced solutions make contact, the precipitation occurs at the miscible interface. The precipitate will locally reduce the permeability by destabilizing the fluid front and inducing viscous fingering [176]. This situation is logical if analyzing Darcy’s law. In numerical terms, a decrement in the permeability will produce the same effect as the increment in viscosity as the pressure gradient is proportional to the factor µ/κ. The physical meaning, however, is completely different, and the instability mechanism is far more complex. Similar studies were done considering studying the effects of increasing the permeability [176, 23, 138, 102]. One major application of reactive viscous fingering is to control processes that are often unstable, like enhanced oil recovery [132, 207, 41, 21] or chromatography [166, 173, 32, 42, 198]. In these cases, studies are conducted in finding suitable reactions that could suppress the instability and improve the fluid displacement. This would be important in many fields of the industry or science. 1.9 The Poly(Acrylic Acid) A very important part of this thesis is to find a coupling between a pH-sensitive material and a pH-changing chemical reaction. In this sense, the most important reactant that will act as a nexus between those different fields is the Poly(Acrylic Acid), or hereafter, PAA. As a key molecule for the development of this work, it is useful to introduce and summarize the most important aspects related to physics and the chemistry of this reagent. 34 DAR´ IO MART´ IN ESCALA VODOPIVEC The PAA is probably one of the most well-known and accessible pH-responsive polymer available. It is currently used in many different fields like the food industry, medicine, cosmetics, pharmaceutics industry, and others [35]. Due to its hydrophilic character, the PAA can absorb a thousand times its weight in water, forming superabsorbent gels [194]. As its name indicates, the PAA is a linear polymer formed from acrylic acid (Figure 1.24). CH2CH C O OH CH2CH C O O−+ H+ Figure 1.24: Chemical model of the acrylic acid dissociation equilibrium in aqueous solution. The PAA structure has a carboxylic group on each monomer unit for every two carbon atoms on the main chain and behaves as a polyelectrolyte in water due to the dissociation of the acid groups [104] (Figure 1.25). CH2CH COOH !n CH2CH COO− !n + nH+ Figure 1.25: Chemical model of the PAA dissociation equilibrium in aqueous solution. By definition, a polyelectrolyte is a polymer composed of macromolecules in which a substantial portion of the constitutional units contains ionizable groups, or both [91]. A conceptual scheme of what a polyelectrolyte is presented in Figure 1.26. Figure 1.26: Schematic of a polyelectrolyte molecule. Specifically for the PAA, the anionic character of the molecule is due to the dissociation of the carboxylic groups into carboxylate ions. 1.9.1 Structure and pH-dependence As all poly(carboxylic acids), the PAA is a weak polyelectrolyte that dissociates in aqueous solutions and its ionization equilibrium is pH-dependent [142]. This can be observed in Figure 1.25. If the H+concentration increases, the pH decreases, favoring the non-ionized form of the molecule where the carboxylic groups are dominant. On contrary, if the pH increases by decreasing the H+concentration, the molecule dissociates favoring the ionized form where the carboxylate ions are predominant. The non-ionized groups facilitate the generation of interchain hydrogen bonds, which compacts the molecule structure (globular form). This compacted structure makes PAA acidic solutions have relatively low viscosity if in diluted solutions. In the ionized form, the negative charges of every dissociated carboxylic group generate a repulsive effect that elongates the molecule into an extended structure (rodlike form). This conformational change strongly affects 35 Chapter 1. Introduction Figure 1.33(b) shows the example presented in Figure 1.33(a) but observed through Schlieren optics. The second case presents a viscous fingering instability produced when a more viscous solution of sucrose is displaced by a less viscous solution of NaCl. As can be seen, the Schlieren technique is not only powerful enough to expose the viscous fingering instability between two colorless solutions but it is also capable to show the density differences between both liquids. This difference in density is appreciated as the black stripes observed inside the fingers [87]. There are more sophisticated arrangements like the z-type Schlieren setup, where the schlieren object is located between two parabolic mirrors instead of a pair of lenses. The light beams pass through the test area of an experimental arrangement aligned in a z-shape form [172]. Figure 1.33(d) presents an example of a cold air stream observed through this technique. As can be appreciated, this technique increases the sharpening and quality of the image but requires more sophisticated equipment and more space disposal. As can be seen, the images obtained through the Schlieren technique are sharper and more detailed than those obtained through the Shadowgraphy. However, as was previously indicated, the Shadowgraph is easier and more cost-effective to implement. 42 PART I: DENSITY FINGERING INSTABILITY DRIVEN BY THE BZ-CHD OSCILLATOR Motivation Oscillating behaviors are common in Nature. They are responsible for a broad range of processes ranging from industrial applications, metabolic cycles, periodic modulations of the environment, human oscillations, and more [134]. Oscillating chemical reactions are well-known examples that attempt to simulate certain behaviors. One of the most studied oscillating reactions is the Belouzov-Zhabotinsky reaction (BZ) [18, 55, 54] which was used as a model to describe dynamic behaviors in natural processes [199, 185, 143, 29]. In this sense, the inherent complexity of the BZ reaction, in addition to its vast and rich behavior, makes it ideal for the reproduction of many natural phenomena [54]. Many processes in Nature often share the common property of taking place in a fluid medium. Hydrodynamic instabilities play an important part in these situations. The spectrum of structures where fluid instabilities play a role is large extending from Nature [200, 171, 111] to the industry [184]. Most recently, hydrodynamic instabilities are considered in several fields such as oil recovery processes [93][41, 132, 44, 207], CO2sequestration [44, 121, 120] among other applications. It is normal to think about systems where chemical and hydrodynamic instabilities exist at the same time and interact synergistically [163]. The knowledge and control of such processes could be essential to understand more complex problems such as marine pollution [125] or chemical gardens[31, 86, 16, 26], among many others. Just recently, and because of the complexity of the problem, these types of coupled systems were considered and analyzed [44, 26, 57, 9, 88, 48]. The forthcoming chapters will introduce a thorough description of a coupled system consisting of a bubble-free oscillating Belousov-Zhabotinsky reaction (BZ-CHD reaction [114, 187, 188, 188, 189]) and a classical interfacial hydrodynamic instability in a vertically oriented Hele-Shaw cell. The system was designed to allow two miscible fluids to interact at the interface, so the reaction takes place only inside the cell. The competition of the two instabilities will be now controlled by two independent parameters: on the one hand, the excitability that deals with the shape and duration of the oscillations and, on the other hand, the initial density gradient across the interface. A detailed quantitative analysis of the effect of each parameter will be presented. The experimental findings will lead to propose a modification of the existing kinetic models for the BZ-CHD reaction [189] that is fully capable to describe the problem in a spatially extended configuration. The results here presented are based on the work published in Escala et al, (2014) ([57]) and Escala et al, (2019) ([61]). Chapter 2 Experimental and Numerical Methods 2.1 Experimental Methods 2.1.1 Hele-Shaw Cell Construction All the density fingering tests were performed in a vertically arranged Hele-Shaw cell [175, 57]. This cell was built utilizing two rectangular plates (20 cm x 10 cm) of scratch-resistant methacrylate (Plexiglass®) separated by a 0.25 mm poly(tetrafluoroethylene) (PTFE) spacer as appeared within Figure 2.1. Two injection holes were set at the upper and lower part of the methacrylate plates (marked with A and B in Figure 2.1(a) respectively). Two additional holes were drilled at both sides and were utilized as fluid outlets (marked with C and D in Figure 2.1(a)). These two holes were horizontally aligned to achieve a planar interface once the reactive solutions were injected. A metallic rectangular frame was put on each side of the cell to fix the device, prevent undesirable harm to the methacrylate and ensure a homogeneous pressure distribution that guarantees an equally spaced gap between the Hele-Shaw cell’s two plates. 2.1.2 Injection Protocol Two solutions containing separated parts of the BZ-CHD reaction were injected utilizing a peristaltic pump (Gilson Minipuls 3) using silicon tubes attached to a chemically resistant Polypropylene connector (made by CPC®, [39]) of 4 mm internal diameter. For this work, two connector models (PMC2201212 and PMC230212) were used. Those two models of connectors were specifically chosen due to their chemical resistance. The procedure to obtain a planar initial condition was performed in two steps similar to Shi et al [175]. First, Solution 1 was injected from the lower hole (B) keeping closed the outlets (C and D) (Figure 2.2(a)). Once the cell was completely filled with this solution, the tube with Solution 2 was connected to the upper injection hole (A) avoiding the inclusion of bubbles. Secondly, once both connectors were in place, outlets C and D were opened and both liquids were injected using a fast flow rate (Figure 2.2(b)). Once the interface between Solutions 1 and 2 was completely planar, the injection was stopped and the outlets closed. This moment was considered the beginning of the experiment (Figure 2.2(c)). The opening of the outlets was controlled using a single chemically resistant PTFE faucet. For all cases studied in the present work, the system was always initially hydrodynamically stable. Thus, the denser fluid (in this case Solution 1) was always injected from the bottom inlet [57]. Chapter 2. Experimental and Numerical Methods Figure 2.1: Schematics of the Hele-Shaw cell and the injection connectors. (a) Frontal and (b) side views of the designed Cell. The Holes named A and B in the scheme (a) are the inlets for Solution 2 and 1 respectively. C and D are the outlets that are only open in order to achieve a planar interface at the center of the reactor. (c) The chemically resistant Polypropylene connectors were used to inject the solutions. The chosen material was suitable for dealing with the products of the BZ-CHD reaction. The location of the PTFE spacer is indicated by the red arrows in (b). 2.1.3 Optical Arrangement The Hele-Shaw cell was placed in an experimental setup described in Figure 2.3. Two different images were recorded for each experiment at the same time. The first one was a “naked eye” view (hereafter chemical view) which was useful to observe all kinds of chemical phenomena like spirals, waves, and spatio-temporal dynamics associated with the chemistry (clearly observed due to the color changes of the catalyst). The second one was an image obtained through the Schlieren technique (hereafter Schlieren view). This Technique is useful to detect variations in the hydrodynamic field that are impossible to observe in the chemical view [172]. The experimental setup was illuminated by using a high-power light-emitting diode (LED) filtered using a variable slit oriented in concordance with the fingers displacement (vertical axis). A first collimator lens was placed in the light pathway close to the LED light source. The collimated light beam passed through the Hele-Shaw cell. A 50/50 beam splitter was placed between the cell and a second collimator lens. The deviated light beam directly impacts a CMOS camera (PixeLink PL-B776U) which recorded the chemical view. The remaining 50% of the light beam passed through a second collimator lens. A knife-edge cutoff filter was placed at the focal point of the second collimator in order to obtain a Schlieren view of the cell which was recorded by a second CMOS camera (PixeLink PL-B776U). A graduated spatial reference was used to obtain quantitative information. 2.1.4 Chemical Recipes and Experimental Designs In the present work, several recipes of the BZ-CHD were used depending on the experimental context. The following section summarizes the reaction and protocols related to each specific part. All solutions presented here were made from reagent grade stocks. More details regarding the stock 48 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 2.2: Schematics of the injection procedure. In (a) the initially empty Hele-Shaw cell was initially filled with Solution 1 from input B. (b) Once filled, Solution 2 was injected from input A and outputs C and D are opened. (c) Both liquids were injected until the planar interface was obtained. At that moment, all the inputs and outputs were closed. Figure 2.3: Experimental setup used for the convective Hele-Shaw system. The arrangement was built by recording two different images from each experiment. A first camera recorded the direct observation of the experiment. This observation was obtained by filming the reflection of the cell through the beam splitter located between the cell and the second collimator lens. The second observation was obtained by using the Schlieren optical technique. The light beam passed through the cell located between two collimator lenses. The knife-edge filter was located at the focal point of the second collimator. 49 Chapter 2. Experimental and Numerical Methods preparations are included in Appendix Section A.1. Both solutions were thermally stabilized at 22 °C by using a thermostatic bath in order to avoid density variations due to temperature changes. The maximum recorded temperature variation between solutions was ∆T=|TS1−TS2|=0.2 °C. All types of thermal artifacts related to the light source can be neglected as the LED source does not produce significant temperature variations. All the experiments were done in a room located in a controlled environment, avoiding significant temperature variations during each experiment. Two main parameters were used for this part of the work: the excitability (ε) and the density difference between solutions 1 and 2 (hereafter density gradient, ∆ρ). The excitability is defined as: ε=[H2SO4]0·[BrO3−]0 [CHD]0 (2.1) this expression was taken from Krinsky [109] and Vanag et al [205]. This value is obtained with the initial concentration of the species of the BZ-CHD reaction. On the other hand, the density gradient is defined as: ∆ρ=ρS1−ρS2(2.2) this value is calculated as the difference between the density of the solution located at the bottom (ρS1), and the density of the fluid located at the upper part of the Hele-Shaw cell (ρS2). This parameter was adopted as a simple measure to evaluate the changes in density due to the chemical variations. All density measurements were done by using a Anton-Paar DMA™ 35 density-meter with an instrumental precision of 0.001 g/cm3. Recipe for the RDC Hele-Shaw System with an independent change of εand ∆ρ.In this case, εand ∆ρwere varied by simultaneously changing [H2SO4]0in both solutions, and [NaSO4]0in Solution 1 respectively. Both parameters were varied up to four different values each obtaining a total number of sixteen experiments. Each experiment was repeated three times. A schematic illustration of the experimental design is presented in Figure 2.4. The recipe and used concentrations are indicated in Table 2.1. Figure 2.4: Schematics of the experimental design used to study independent changes of εand ∆ρ. The excitability of the system was varied by changing [H2SO4]0simultaneously in solutions 1 and 2, and it was calculated as indicated in Eq. (2.1). The density gradient was varied by changing [Na2SO4]0in Solution 1. This value was calculated as indicated in Eq. (2.2) by measuring the densities of solutions 1 and 2. 50 DAR´ IO MART´ IN ESCALA VODOPIVEC Recipe for Solution 1 Recipe for Solution 2 Species Concentration (M) Species Concentration (M) H2SO40.847, 1.693, 2.540, 3.387 H2SO40.847, 1.693, 2.540, 3.387 CHD 0.291 BrO3–0.142 Na2SO40.130, 0.155, 0.205, 0.230 Na2SO40 [Fe(phen)3]2+ 0.4x10−3[Fe(phen)3]3+ 0.4x10−3 Table 2.1: Base BZ-CHD recipe used in the RDC experiments where εand ∆ρvaried independently. Tables 2.3(a,b) show the values of εand the average ∆ρcalculated by changing [H2SO4]0and [Na2SO4]0respectively. These calculations were done by using the concentrations indicated in Table 2.1. H2SO4(M) ε(M) 0.847 0.413 1.693 0.826 2.540 1.239 3.387 1.653 (a) Na2SO4(M) ∆ρ(g/cm3) 0.130 0.002 ±0.001 0.155 0.004 ±0.001 0.205 0.009 ±0.001 0.230 0.011 ±0.001 (b) Table 2.3: Values for εand ∆ρcalculated from Eqs. (2.1) and (2.2) respectively. (a) Values of εobtained by changing [H2SO4]0. (b) Values of ∆ρobtained by changing [Na2SO4]0. The density values are indicated as the average value between three replicas ±the standard deviation. Recipe for the RDC Hele-Shaw System with a coupled change of εand ∆ρ.This set of experiments were made by varying the NaBrO3concentration in Solution 2. Thus, this not only changed the system excitability but also the density gradient. In addition, it is important to remark that the following recipe includes sodium bromide (NaBr) in its formulation, the H2SO4concentration in each solution is not equal and it does not includes Na2SO4. The addition of sodium sulfate was not necessary 51 Chapter 2. Experimental and Numerical Methods introduced in Section 1.47 to describe the dynamics of a chemically induced density fingering instability, were adapted and numerically solved by the software. The reactive terms were taken from the Skeleton model presented in the table 2.9 and were used together with a stiff chemistry solver. The density function, the permeability, and the kinetic laws of the skeleton model were implemented through user defined functions (UDF). The calculation domain (D) and the initial reagent conditions are indicated in the schematic of the Figure 2.6. A squared mapped mesh of 181 elements in the x-direction and 181 elements in the y-direction was used as the numerical grid. The gravity field was set to a value of 9.81 m2/salong the y-axis. As will be described in further chapters, the base model equations for the RDC simulations have been modified from the classical formulation to include the chemical species involved in the instability mechanism, which is the quinhydrone complex ([Q·H2Q]). Even though the development of such modification is extensively detailed in Chapter 5, some information regarding the configuration of the RDC model must be included in this section to comply with the thesis structure. In this sense, the boundary conditions for this particular case were set as zero diffusive flux for all the species except for the quinhydrone, which was set as [Q·H2Q]0= 0 at the lower boundary for numerical stability reasons. This is also indicated in Figure 2.6 Figure 2.6: 2D numerical domain used to simulate the RDC model. The mesh consists of 181 elements in the horizontal direction and 181 elements in the vertical direction. Zero diffusive flux boundary conditions are set at the boundaries except for the [Q·H2Q]. The chemical species were spatially located similar to their experimental counterpart by the following piecewise function: Cai(x,y,0) = ([Ca]0(x,y)(ξr(x,y)+1)y<L/2 0y⩾L/2 Cbi(x,y,0) = (0y<L/2 [Cb]0(x,y)(ξr(x,y)+1)y⩾L/2 [H2SO4]0(x,y,0) = k,∀(x,y)∈D (2.4) 58 DAR´ IO MART´ IN ESCALA VODOPIVEC where [Ca]0are: [CHD]0, [Fe(phen)3]2+ 0, [Na2SO4]0and [Cb]0are: [NaBrO3]0and [Fe(phen)3]3+ 0.r(x,y) is a normally distributed random function with amplitude ξ= 10−2. [H2SO4]0was set constant for the entire domain. The pressure field was calculated by using a second-order upwind scheme, while the chemical species were calculated by using a first-order upwind scheme. The time step was automatically controlled by the software based on an adaptive algorithm using the first-order upwind discretization [11, 12]. All simulations were calculated for a final time of 200 s. All the parameters used for RDC simulations are listed in Table 2.10. Parameter Value Dimension Unit Reference/Notes [BrO3–]00.142 M experimental [CHD]00.291 M experimental [Fe(phen)3]2+ 04×−4M experimental [Fe(phen)3]3+ 04×−4M experimental αBrO3−0.114×10−3M−1experimental (App. D) αCHD 0.019×10−3M−1experimental (App. D) α[Fe(phen)3]2+/3+ 0 0.264×10−3M−1experimental (App. D) αNa2SO40.121×10−3M−1experimental (App. D) αH+0.013×10−3M−1ad hoc [Na2SO4]00.108, 0.200 M experimental/adjusted [H+]03, 15 M model ρ01.000 g cm−3experimental µ0.001 Pa.s experimental P00 Pa L 0.01 m Di1×10−8m2s−1all species, except for Q·H2Q DQ·H2Q1×10−9m2s−1estimated a2.5×10−4m experimental κ05.208×10−9m2experimental: a2/12 RK0.75 experimental/adjusted RMMQ·H2Q218.2 g mol−1reference [214] ρS1.402 g cm−3reference [214] φ1 reference [197] γ100 ad hoc VT2.5×10−8m3calculated from L2a krQ·H2Q5×10−2M−1s−1estimated Table 2.10: Parameters used in the RDC simulations. All values are expressed in the International System of Units (SI) excepting concentrations that are expressed in Molarity. 59 Chapter 3 Experimental Results Abstract: This chapter will show a complete experimental characterization of the coupling between the BZ-CHD and the density fingering instability. Results are presented in both, descriptive and quantitative manner, showing the effect of the most important factors involved in the dynamics of the system. The conclusions obtained from the experimental observations will be fundamental for the understanding of the coupling mechanism and will facilitate the subsequent development of a convective model. 3.1 General System Overview The characteristic behavior of the experimental system described in Section 2.1 is presented in Figure 3.1. These results were obtained for a reference case where ε= 1.653 M and ∆ρ= 0.004 g/cm3, which corresponds to initial concentrations of [Na2SO4]0= 0.155 M and [H2SO4]0= 3.387 M. Figure 3.1: (a) Chemical and (b) Schlieren views of an experiment with ε= 1.653 M and ∆ρ= 0.004 g/cm3. The experiment started with an initially stable planar interface. Solution 1 was located at the bottom and solution 2 is at the upper part of the image. At t= 27 min, traveling waves were observed moving through the interface from the left to the right side of the reactor. At t= 47 min, fingering instability was observed in the Schlieren view. The effect of the fingers on the interface was visible in the chemical view as the interface became deformed. The traveling waves were also affected by the finger onset as they travel through the fingers. In the last frame at t= 73 min, a strong reddish precipitate appears at the finger contour (observed in both views). (c) Shows the spatio-temporal drift of the representative finger marked with a red dashed oval in panel (b) frame 47 min. In the frame are indicated the vertical and horizontal displacement distances. Figures (a) and (b) were adapted from [61] Chapter 3. Experimental Results Figure 3.1(a) shows a direct observation of the recorded experiment (chemical view) and Figure 3.1(b) shows the same experiment but observed through the Schlieren technique. Four frames are presented corresponding to consecutive times. The initially stable condition is shown at t= 0 min and was obtained by using the method exposed in Figure 2.2. Solution 1 (higher density) was situated at the bottom and solution 2 (lower density) was situated at the upper part of the Helle-Shaw cell. The reaction started in the vicinity of the interface where all chemicals of the BZ-CHD reaction were mixed by diffusive processes. In descriptive terms, starting from t= 27 min, it can be observed how chemical waves pass through the interface in the chemical view. No specific pattern was observed in the wave dynamic as they move independently from left to right or the opposite depending on the experiment. In this particular case, the waves were moving from left to right. In other experiments, wave collision can be also observed at the center of the image. At this stage, no significant motion in the fluid was observed as can be observed in the Schlieren view. This was expected due to the initially stable condition. At t= 47 min, the beginning of the fingering instability was observed in the Schlieren view. The deformation in the interface observed in the chemical view was directly produced by the finger onset. The fingering instability is shown fully developed at t= 73 min. From the chemical view is possible to observe how the initial interface moved from its original position rising to the upper part of the reactor. In addition, a reddish precipitate that emerged at the interface between both solutions was observed. This precipitate was deposited in the interface adopting the shape of the fingers. Another interesting phenomenon is the finger displacement through both vertical and horizontal directions. Commonly, finger patterns originated from buoyancy-driven instabilities displaces vertically due to the gravity action. However, it was observed in some cases that fingers also moved in the horizontal direction. Figure 3.1(c) shows the drift of a sample finger along with the horizontal and vertical directions. The finger lateral excursion and vertical displacement were measured by isolating the movement of an individual finger marked with a red dashed oval in Fig. 3.1(b) frame 3. This image was obtained by averaging several temporal snapshots into one single frame. Both displacements are indicated in millimeters. 3.2 Descriptive Analysis of the Effect of ∆ρand ε To have a better understanding of the effect of the density gradient between the two solutions and the excitability of the system, several experiments were performed by varying the excitability and the density gradient independently. These results were obtained using the experimental design presented in Table 2.3. In Figure 3.2, an extended summary between all the experimental cases is presented. All frames are snapshots taken 1 h after the beginning of each experiment. Horizontal and vertical axes represent the density gradient ∆ρ(Eq. (2.2)) and the excitability ε(Eq. (2.1)) respectively. Each point in this diagram shows two observations, the Schlieren view and the chemical view for each experiment. By analyzing the Schlieren images, it is possible to observe an increment in the stability of the interface in line with the increment in the density gradient. This was expected as the main stability factor in buoyancy-driven instabilities is the density gradient (Section 1.8). For lower values of the excitability (ε= 0.413 M and 0.826 M respectively), the instability was completely suppressed once the density gradient reached ∆ρ= 0.009 g/cm3. For larger values of the excitability (ε= 1.239 M and 1.653 M), the increment in the density gradient did not completely suppress the instability but damped the amplitude of the fingers down. Changes made by increasing the system excitability produced four remarkable effects. Firstly, the interface got crossed by the fingers for lower density gradients. This effect became less significant as the density gradient was increased, which is logical as the stability of the system was increased proportionally to ∆ρ. Secondly, the chemical wavelength decreased and the wave velocity increased. 62 DAR´ IO MART´ IN ESCALA VODOPIVEC This is typically a characteristic in the chemistry of the BZ-CHD oscillator as has been demonstrated in previous works [190, 189]. Thirdly, the amount of precipitate increased with the excitability while the finger onset times were reduced inversely proportional to the excitability. Finally, the finger wavelength decreased. Figure 3.2: Qualitative comparison of all the experiments realized. All snapshots were taken around 1 h after the beginning of the experiment. Each pair (ε,∆ρ) shows two experimental observations of the same experiment at the same time. Both, the Schlieren view (left side) and the chemical view (right side) are plotted for each pair (ε, ∆ρ). The system got stabilized by increasing ∆ρ. This strongly affects the hydrodynamic by delaying the finger onset and stabilizing the initial interface. By increasing ε, the finger, chemical, and precipitate onset times were decreased, while the hydrodynamic and chemical wavelength also decreased. This Figure was taken from Escala et al [61]. 3.2.1 Measuring Observables In contrast to the descriptive analysis previously made, several macroscopic observables were calculated to obtain quantitative measurements of the system dynamics. In the present work, four main observables were chosen, two of them used for chemistry characterization and the other two for hydrodynamics characterization. Formally, the chemical oscillations period (TC) and the chemical reaction induction time tind−C(this is, the time elapsed from the start of the experiment till the first waves are observed) were chosen to measure the effect of the chemistry in the whole process. On the other 63 Chapter 3. Experimental Results hand, the finger wavelength, λH, and the hydrodynamic induction time tind−H(considered as the time elapsed from the beginning of the experiment till the fingers start developing) were the observables used to characterize the hydrodynamics. The observables were directly measured from the space-time plots (STP) obtained from the experimental observation and using the methodology indicated in Appendix Section C.1. Figures 3.3-3.4 plots the variation of the macroscopic observables previously defined as a function of εfor each ∆ρanalyzed. All values are presented as the average value over all the realizations and error bars show the standard deviation in the measurements. The value of ∆ρused for each case is indicated in the legend in order to facilitate graphic comprehension. As can be observed in Figure 3.3(a), variations in the excitability produced a remarkable effect on the chemical induction time. More precisely, tind−Cgot sensibly reduced for high values of ε. The same effect occurred for every ∆ρstudied. The density gradient did not produce any significant change on this observable in any case. Furthermore, except for the lower excitabilities, the deviations in the measurements were relatively small indicating that this tind−Cis a purely chemical characteristic that is independent of the hydrodynamic condition. Figure 3.3: (a) Chemical induction time tind−Cand (b) Chemical period TCas a function of εand ∆ρ. Figure adapted from Escala et al [61]. On the other hand, the variation of TCas a function of εand ∆ρis shown in Figure 3.3(b). In a similar manner to tind−C, the chemical period TCwas strongly affected by changes in εbut no effect was observed by changing ∆ρ. In this case, the largest dispersion was observed for ε= 0.413 M (lower excitability case), where the average TCvalue ranges between 9.86 min and 10.81 min. The dispersion dramatically reduces for ε>0.413 M and the period decreases uniformly for each ∆ρ. For ε= 1.653 M the minimum average TCvalue ranges from 27 up to 31.8 s. Regarding the hydrodynamics, Figure 3.4(a,b) shows the variation of tind−Hand λHrespectively. In the first case, the system response to variations in εand ∆ρwas more complex compared to the previous observables. For ∆ρ= 0.002 g/cm3and 0.004 g/cm3, the increments in εdid not show any statistically significant variation considering the dispersion of the measured values. However, the behavior of the system changed abruptly for ∆ρ= 0.009 g/cm3and 0.011 g/cm3. For these two cases, the value corresponding to ε= 0.413 M was not included in the figure as no fingering instability was observed during the experimental time. For ε>0.413 M, tind−Hdecreased by increasing the excitabilty in all cases but no significant variations were observed by changing the density gradient. These results were consistent with previous works [197, 9] and shown how the chemistry can affect hydrodynamics. 64 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 3.4: (a)Hydrodynamic induction time tind−Hand (b) Hydrodynamic wavelength λHas a function of εand ∆ρ. Values of tind−Hcorresponding to ε= 0.413 M for ∆ρ= 0.009 and 0.011 g/cm3were not plotted as no fingering instability was observed during such experiments. Figure adapted from Escala et al [61]. Finally, as can be appreciated in Figure 3.4(b), no statistically significant variations of λHwere observed for most εvalues, except for larger densities, ∆ρ= 0.009 g/cm3and 0.011 g/cm3, where finger wavelength decreased slightly inversely to the excitability, ranging between 0.14 cm up to 0.26 cm. 3.3 Coupled ∆ρand εVariation by Changing [BrO3–]0 Until now, only the effects produced by independent changes in the excitability and the density gradient were studied showing the major role played by the chemistry in the system behavior. However, on the basis of Equation (2.1), the excitability can be modified not only by changing the acid concentration but also by changing the initial concentrations of bromate and CHD. This section will be focused on the effect of the bromate in the dynamics of the system. However, as this species is in the upper layer, it is not possible to add a heavy salt to the solution (such as Na2SO4 in the previous case) for obtaining independent changes in density and excitability. Thus, both ∆ρand ε will be affected simultaneously by changing [BrO3–]0. In this case, the hydrodynamic field was observed by using the shadowgraph technique instead of the Schlieren technique, which was described in Section 1.101. More information regarding the experimental protocol, recipes, and general details can be consulted in Section 2.1.4. By analyzing the dependence of the system with the excitability, different behaviors were found in concordance to the results presented in Section 3.2. The chemical and the hydrodynamic induction times were much larger for lower values of ε(Figure 3.5(a), compared with the middle and high excitability cases. Likewise to Figure 3.1, phenomena as chemical waves with a large wavelength and the displacement of the interface were observed as well. Convective fingers were observed about 200 min from the beginning of the experiment. Similar to the lower εcase of the results presented in Figure 3.2, the fingers did not move beyond the initial interface, and in the same way, they grew in the vertical direction and moved along the horizontal direction. For larger excitabilities (Figure 3.5(b,c)), the system response was also similar to their counterpart in Figure 3.2. The hydrodynamic instability was 1This technique was used due to resource constraints at the moment in which the experiments were conducted. 65 Chapter 3. Experimental Results Figure 3.5: Description of system behavior for (a) low (ε= 0.334 M), (b) middle (ε= 0.557 M), and (c) high excitabilities (ε= 0.668 M). The chemical view is indicated in the upper rows and shadowgraph view at lower rows of each panel. The system evolution showed a comparable behavior with the case presented in the previous section for different increments in ε. The white dashed line indicated in (b) represents the horizontal cut used to build the space-time plots of Figures 3.8 and 3.6. All frames were taken at the experimental times indicated below each panel. The results are presented using a false-color palette to facilitate the observation. Figure adapted from Escala et al [57]. triggered as soon as chemical waves appeared and both chemical and hydrodynamic structures exhibited 66 DAR´ IO MART´ IN ESCALA VODOPIVEC analogous wavelength. In addition to the traveling waves, spirals were also observed interacting one each other (Figure 3.5(c)). Those spirals were originated from the fingertips showing the degree of interaction between both, the chemical and the hydrodynamic process. Figure 3.6: Space-time plots created by overlayering the chemical and the hydrodynamic patterns for the cases shown in Figure 3.5. (a) ε= 0.334 M, (b) ε= 0.557 M and (c) ε= 0.668 M. Horizontal and vertical bright stripes feature the chemical waves and the hydrodynamic fingers dynamics, respectively. As can be noted, as the chemical excitability is increased the convective dynamics of fingers become more complicated as a result of a complex chemical forcing. Figure adapted from Escala et al [57] A supplementary analysis of these processes is also presented in Figure 3.6, where the transition from simple to complex chemo-hydrodynamic behaviors was characterized by comparing the space-time plots of the cases presented in Figure 3.5. Here the superimposition of the chemical and the shadowgraph views make it possible to appreciate the correlation between the chemical and hydrodynamic patterns previously seen in Figure 3.5. Figure 3.6(a) describes the case for ε= 0.334 M where a long-wavelength train of waves moved from left to right. For Figure 3.6(b) and (c), the excitability of the system was increased from ε= 0.557 M and 0.668 M respectively. For these cases, chemical waves showed complicated patterns with a competition between target and spiral waves coming from different directions which collided in the mixing zone. As a result, the hydrodynamic pattern, closely following the chemical dynamics, exhibited an intricate behavior that changes with time. 67 Chapter 4. Detailed Chemical Analysis in the finger onset time. The figure compares the tind−Hwith two extreme values, both indicated with grey dashed lines in the plot. The line on the bottom side indicates the induction time for a catalyzed reference experiment. This value corresponds with the experiment where no salt was added. On the upper side is indicated the onset time of a non-catalyzed case. This value was obtained from an experiment without ferroin. As is possible to see, tind−Happroached the non-catalyzed time for the highest NaCl concentration. The inverse situation was observed when the salt concentration was one order less. The total increment in the induction time was up to 620 %. Aside from the delay in the induction time and the inhibition of the spatio-temporal patterns, no other significant effect was observed in the system dynamics. Figure 4.1: Effect of the addition of sodium chloride in the BZ-CHD reaction. (a) Frames taken from the chemical view for NaCl = 0.0040 M and (b) NaCl = 0.005 M. (c) Hydrodynamic induction time (tind−H) variation due to the addition of NaCl studied for [NaCl]0= 0.001 M, 0.002 M, 0.003 M, 0.004 M 0.005 M, 0.007 M and 0.01 M respectively. The remaining reagents were kept constant as indicated in Table 2.4. For this particular case, [BrO3–]0= 0.095 M which corresponds to ε= 0.557 M. The experimental frames were colored to facilitate the visualization. 4.1.2 Experiments Without CHD The previous results indicated that fingering instability occurs with the uncatalyzed reaction. On the other hand, the presence of the organic substrate is fundamental for obtaining chemical oscillations [113]. Therefore, it is necessary to study the role played by CHD in the instability. Thus, a control experiment where the CHD was removed from the system is presented in Figure 4.2. For this case, the base recipe that was used is presented in Table 2.4. The removal of CHD was compensated with the addition of Na2SO4in Solution 1 to obtain a comparable density jump. Figures 4.2(a,b) show the chemical and shadowgraph views respectively. As can be seen, both cases show a diffusive front in which the ferroin was oxidized into ferriin. Neither chemical nor fingering patterns were observed during the whole experiment. This result demonstrated that CHD is a key species in the instability mechanism. Also, it was useful to discard the hypothesis that the instability may be generated by the change in the oxidation state of the catalyst [26]. 74 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 4.2: (a) Chemical view row and (b) shadowgraph view row of an experiment where [BrO3–]0= 0.095 M (which corresponds to ε= 0.557 M and ∆ρ= 0.002 g/cm3) where CHD was removed from the recipe. Note that neither the instability nor pattern-formation is observed. The experimental frames were colored to facilitate the observation. 4.2 UV-Vis Spectroscopy The system was studied in a fully stirred batch reactor by using UV-Vis spectroscopy techniques as indicated in the corresponding Section 2.1.5. The recipe used for this case is indicated in Table 2.6. The chemical behavior was characterized similarly as the convective system, where the temporal dynamic and the chemical observables were calculated. The dynamics of the system is presented in Figure 4.3(a) for three different cases: ε= 0.207, 0.826, and 1.653 M. These values were representatives of three different excitability conditions, as low, middle, and high excitabilities respectively. For the lower case, no oscillations were observed but only a single shift from the oxidized (Fe+3) to the reduced (Fe+2) state of the catalyst. For the middle and the high excitable cases, oscillations were observed for a considerable range of time. Those oscillations are shown in Figure 4.3(b) where a deep view of the oscillatory region corresponding to the high excitability case is presented in the inset (c) of the figure. In addition, a signal saturation region was observed (Figure 4.3(a)). This saturation occurred in experiments with medium and high excitabilities whereas the absorbance signal diminished after the shift in the oxidation state for ε= 0.207 M. From the temporal dynamics is possible to observe the effect of the excitability on the overall system behavior. By increasing the excitability, the induction time was reduced. This was particularly noticeable for the low excitability case where the induction time was much larger compared with the other two cases. Regarding the oscillations, the increment in the excitability reduced the total amplitude of the oscillations, showing a progressive increment in the amplitude in time, whereas for the middle excitability case, the amplitude remained constant for all the oscillatory region. Also, one last oscillation was observed for the middle and high excitability cases. All these chemical characteristics were also quantitatively analyzed by measuring the chemical 75 Chapter 4. Detailed Chemical Analysis Figure 4.3: (a) Dynamic characterization of the batch system for ε= 0.207, 0.826, and 1.653 M by using UV-Vis spectroscopy. All spectra were obtained at a fixed wavelength of 510 nm. The absorbance values were converted into the molar concentration considering an absorptive molar coefficient ξmax = 11000 L/(mol.cm)[68]. Only three cases are shown to facilitate the analysis. (b) spectrum corresponding to the high excitability case where the oscillations region is shown in the figure inset (c) for a range of time of 100 s. The saturation region is indicated in the blue dashed square. This Figure was adapted from Escala et al [61] period (TC) and induction time (tind−C). All the information is presented in Figure 4.4 and was obtained by direct measurement of the spectroscopy spectra done for all the cases indicated in Table 2.6. The induction time characterization is shown in Figure 4.4(a). As can be seen, the induction time got significantly reduced conforming the excitability was increased. It is interesting to appreciate that not only the trend of tind−Cis similar to the convective case (Figure 3.3(a), but also the induction time values were approximately of the same order. This also confirmed the major relevance of the chemistry in the RDC case. The chemical period was also characterized and it is presented in Figure 4.4(b). For this case, the period of the oscillations was in the order of seconds rather than minutes as observed in the RDC case (Fig. 3.3(b)). However, the trend in the period was comparable with the convective case showing a significant decrement due to the excitability increment. Oscillations were observed from ε⩾0.619 M. 76 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 4.4: Chemical observables characterization obtained for a broad range of excitabilities as indicated in Table 2.7. (a) Chemical induction time (tind−C) and (b) chemical period (TC), for the batch system. For (b) cases ε= 0.206 and 0.413 M were not included in the plot as no oscillations were observed. 4.3 Precipitate Formation The chemical analysis done confirmed the main role played by chemistry in the convective dynamic. The chemical observables of the fully stirred system qualitatively conserved the main characteristics of the spatially extended system. The control experiments demonstrated the strong influence of the CHD in the instability development and the effect of removing the catalyst. Finally, similar dynamics were observed when changing the excitability by varying [H2SO4]0and [BrO3–]0. All these results suggested the possibility that the instability was produced by a species derived from the organic substrate (CHD). From the UV-vis spectra, it was observed a signal saturation for medium and larger excitabilities (Fig. 4.3). On the other hand, the reddish precipitate observed in fingering instability also occurred for medium and larger excitabilities (Fig. 3.2). Therefore, these results suggested that the species generated in the saturation region would be related to the instability mechanism. Several experiments were conducted to understand how the signal saturation was produced. Figure 4.5(a) shows a set of frames of the BZ-CHD reaction done in a stirred assay tube which is directly comparable with a spectroscopic experiment. The initial concentrations used for this case were the same as used in the experiment presented in Figure 4.3(b) (ε= 1.653 M), which corresponds to a high excitability case. The exact moment when two oscillations occurred is indicated between 849.6-852.4 s and 1216-1227 s. Initially, the ferroin was in an oxidized state (ferriin) becoming in a reduced state from 1600 s. From that time, no more oscillations were observed. In an instant between 2520 and 2880 s, the solution suddenly changed its coloration becoming blackish and turbid. This corresponds with the signal saturation observed in the UV-Vis spectrum (Fig. 4.3(b)). It is also interesting to observe how the coloration of the solution changed in time, starting from a deep blue coloration (ferriin), acquiring then a greenish tone, then red to finally get colored into black. This suggested the idea that some secondary species or products also affected the typical coloration of the redox indicator. A similar situation was observed in the uncatalyzed reaction which is presented in Figure 4.5(b). For this case, the solution was initially colorless. As time progresses, the solution acquired a yellowish coloration. This coloration became stronger, turning into orange around 2880 s. Like the catalyzed experiment, the solution became turbid in a time-lapse between 2880 and 3600 s. The 77 Chapter 4. Detailed Chemical Analysis Figure 4.5: Fully stirred experiments for (a) catalyzed BZ-CHD reaction and (b) uncatalyzed BZ-CHD reaction both for ε= 1.653 M. The remaining reagents were kept as indicated in Table 2.6. Figure adapted from Escala et al [61] yellowish coloration in the uncatalyzed experiment explained why the catalyzed case acquired a greenish coloration. This effect was produced by the mixture between the blue coloration of the oxidized state of the ferroin and the yellow coloration of the observed in the uncatalyzed experiment. The turbid coloration observed in both cases was due to the emergence of a heavy precipitate suddenly originated at the end of each experiment. In particular, the precipitate onset showed some delay in the uncatalyzed experiment compared to the catalyzed one. This also led to the suspect that the rate of production of the precipitate is catalyzed by the ferroin. This precipitate was also observed in the convective experiments as was exposed in Section 3.1. Additionally, all these facts were also confirmed by measuring the color intensity variation as a function of time. These values were taken from both experimental recordings by studying the temporal profile of a sample region indicated with the withe dashed line in Figure 4.5(a). The results, presented in Figure 4.6, show the similarities between the results obtained from a completely independent method and those obtained from spectroscopy. The oscillations in the catalyzed experiment (Fig. 4.6(a,c)) and the precipitate dynamics of the uncatalyzed experiment (Fig. 4.6(b)), were observed with this method. These results demonstrated, in qualitative terms, that the signal saturation was produced by the precipitate formation. Regarding the convective system, as previously shown in Figures 3.1, 3.2, and 3.9, the precipitate was also observed at the end of the experiments. In Figure 4.7, a detailed observation of an uncatalyzed experiment shows the precipitate formation (Fig. 4.7(b)) for a middle excitability case (ε= 1.239 M). In Figure 4.7(a) the fingering pattern was observed due to the change in coloration produced by the localized generation of the precipitate. In Figure 4.7(c), it was also possible to appreciate how the precipitate moved downwards through the Hele-Shaw cell. Figure 4.7(d) shows an enlarged image of the precipitate particles sinking. Finally, as explained in the Introduction (Section 1.8.3), some buoyancy-driven chemo-hydrodynamic instabilities may be produced by the emergence of heavy product at the interface between both liquids. It is due to this observation that the species involved in the precipitate formation may be responsible for the development of the instability. It is then important to fully characterize not only its structure but the possible mechanism of generation of this compound. 78 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 4.6: Dynamics of the (a) catalyzed and (b) uncatalyzed BZ-CHD reactions obtained by direct measurement of the results presented in Fig. 4.5. The plots were constructed by analyzing the changes in the color intensity of the aforementioned experiments. (a) Dynamics of the catalyzed BZ-CHD reaction. (b) Dynamics of the uncatalyzed BZ-CHD reaction. (c) Zoom of the oscillatory region showed in (a). In both cases, this method reproduced the saturation region observed through spectroscopic techniques. Figure 4.7: Frames of an uncatalyzed experiment with middle acid concentration. (a) An enhanced image of the fingering formation obtained by direct observation. (b, c) Precipitate formation after the finger onset. (d) Zoomed image of the precipitate moving downward the Hele-Shaw cell. ([CHD]0= 0.291 M, [BrO3–]0= 0.142 M, and [H2SO4]0= 2.540 M). This Figure was taken from Escala et al [61] 79 Chapter 4. Detailed Chemical Analysis 4.4 Nuclear Magnetic Resonance (NMR) Spectroscopy The next step done to characterize the nature of the precipitate was to analyze the compound by a more sophisticated spectroscopic tool. The nuclear magnetic resonance (NMR) was then used to elucidate the chemical structure of the precipitate [24]. The first task was to extract the precipitate from the reaction beaker. This has been done by filtering and drying the BZ-CHD solution as explained in Section 2.1.6. An image of the extracted precipitate is shown in Figure 4.8. Figure 4.8: Precipitate extracted from the homogeneously stirred BZ-CHD reaction. The precipitate was obtained by filtering and drying as explained in Section 2.1.6. For this analysis both, Hydrogen-1 (1H) and Carbon-13 (13C) NMR were performed by using two different solvents: deuterated dimethyl sulfoxide (DMSO-d6) and deuterated methanol (MetOH-d4). The respective results are presented in Figures 4.9-4.10. For the DMSO-d6, Figure 4.9 shows the 13C (Figure 4.9(a)) and 1H (Figure 4.9(b)) spectra. The measured chemical shifts of each peak were marked on top of each one and the values can be compared with the theoretical values obtained with the software Mestre-C for the molecules sketched on the left of each plot. Four peaks were obtained associated with two different types of interactions (C-C and C-O) in each molecule (Figure 4.9(a)). The peaks related to 1,4-hydroquinone (H2Q) were located at 150.14 ppm and 116.10 ppm showing a good agreement with the expected values at 151.42 ppm and 117.45 ppm. The two other peaks at 188.12 ppm and 136.98 ppm correspond with the expected values for the 1,4-benzoquinone (Q) (theoretical values are 187.00 ppm and 135.58 ppm). On the other hand, in the 1H spectrum shown in Figure 4.9(b) two peaks were observed for each C-H bond in each molecule, and a collection of smaller peaks between 8.42 ppm and 8.33 ppm associated with the O-H bond. Again, the measured values for the chemical shifts (6.83 ppm for Q and 6.53 ppm for H2Q) were in good agreement with the expected ones (6.90 ppm and 6.66 ppm respectively). Results were similar for the experiments done with MetOH-d4. The results are presented in Figure 4.10(a) for13C and 4.10(b) for 1H. In the 13C case, the theoretical shifts predicted for Q were 135.58 ppm for and 187.01 ppm, where the experimental ones obtained were 136.21 and 187.32 ppm respectively. For H2Q, the theoretical shifts were 151.74 and 117.50 ppm where the experimental ones obtained were 115.39 and 149.81 ppm. For the 1H case, the theoretical displacement for the H-C interaction in Q was 6.91 ppm compared to 6.77 ppm for the experimental shift. For H2Q, the chemical shifts for the H-C and H-O interactions were 6.65 and 8.23 ppm for the theoretical and 6.61 and 8.29 ppm for the experimental 80 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure 4.9: NMR spectra with DMSO-d6as solvent for (a) 13C and (b) 1H. In both figures, the experimental shifts are compared with the theoretical predictions indicated over the schematic molecules. As can be seen, the experimental chemical shifts agreed with the theoretical ones. cases, respectively. In both cases, the overall agreement between the experimental measurements and the theoretical estimations was very good with a maximum error of less than 0.1 ppm. Figure 4.10: NMR spectra results for MetOH-d4solvent for (a) 13C and (b) 1H. Similar to the results presented in Figure 4.9, the theoretical and experimental shifts agreed significantly. All these results strongly suggested that the precipitate was composed of Q and H2Q. Nevertheless, both species are key chemical intermediaries of the BZ-CHD reaction that play a major role in the dynamics of reaction as detailed in Szalai et al [189]. However, when Q and H2Q reach certain levels of concentration, a non-soluble chemical complex, known as quinhydrone (or Benzoquinhydrone - Q·H2Q) can be formed by the electronic attraction between the aromatic rings of Q and H2Q [37]. A representation of the molecular structure of the complex quinhydrone is presented in Figure 4.11. As was detailed in Section 1.8.3, many instabilities such as those induced by reactions of type A + B →C, occur due to the differential density between product C and reactants A and B. However, in this case the reaction is not elementary and several reactants and products are involved. The precipitate quinhydrone can be considered a denser product capable to induce the fingering instability with a similar mechanism to those simpler cases. Nevertheless, due to the complexity of the reaction, the system has now many different control points that can add richness to the phenomenon. 81 Chapter 4. Detailed Chemical Analysis Figure 4.11: Chemical structure representation of the complex quinhydrone. 4.5 Chapter Discussion In the present chapter, many control experiments were presented in order to elucidate the mechanism of the fingering instability produced by the BZ-CHD reaction. The detailed chemical analysis was useful to isolate every possible control point of the system establishing the main species involved in the observed phenomena. The addition of sodium chloride inhibited the action of the catalyst leading the system into a uncatalyzed state. This experiments showed that fingering instability occurred even without the catalyst. This suggested that the core mechanism of the BZ-CHD reaction was responsible to induce the instability. The experiments done without CHD showed a single chemical front produced by the oxidation of the ferroin into ferriin. In such experiments, no fingering was observed. This suggested that the CHD (or its derivatives) is a key species in the development of the instability. This result also confirmed the experiments done by adding NaCl. As the ferroin/ferriin switch was no capable to induce the instability, this completely demonstrated that such species was not involved in the fingering phenomenon. The results obtained from the UV-Vis spectra demonstrated the qualitative equivalence between the chemical observables obtained from the convective experiments with those obtained from the batch system. In both cases, the increment of the excitability decreased the chemical induction times and period, demonstrating the role played by chemistry in the convective system. The extensive chemical analysis allowed to correlate the precipitate observed in the Hele-Shaw cell with the one observed in a batch reactor. The NMR spectroscopy was used as a the definitive tool to unveil the chemical structure of the precipitate, the quinhydrone (Q·H2Q) complex. This species was generated by the complexation of two components already present in the BZ-CHD reaction mixture, Q and H2Q. Both are key reaction intermediaries obtained from the oxidation of the CHD by the BrO3–and the catalyst as shown in previous work done by Szalai et al [189]. The yellow coloration observed in the uncatalyzed convective system was also indicative of the presence of 1,4-benzoquinone (Q) in the medium, which resulted in an additional confirmation of the proposed mechanism. All the information obtained was crucial to develop a suitable model capable to reproduce the behavior and dynamics observed in the convective system. This topic will be addressed in the following chapter. 82 Chapter 5 Numerical Results Abstract: The previous chapter showed that the precipitate observed in both, the convective and batch systems was quinhydrone. This compound is generated by the complexation of Q and H2Q. The results suggested that the instability may be produced by this species through a classical A + B →C like mechanism. To prove this hypothesis, non-linear numerical simulations were done to obtain comparable results. In this context, the present chapter will study the equivalence between the homogeneous system and the available kinetics models. Once known the optimum set of parameters that better represent the experimental results, the BZ-CHD kinetic models introduced by Szalai et al [188, 187, 189, 190] will be adapted to a spatially extended configuration to simulate the reaction-diffusion and reaction-diffusion-convection systems. All these numerical models will be used as the definitive tool to demonstrate the mechanism of the instability. The main quantitative results presented in the experimental section were recalculated from the simulations and compared with the experimental values to show the agreement between them. 5.1 Equivalence between Experiments and Reaction Models 5.1.1 Qualitative Comparison The results obtained by spectroscopy were directly compared with the extended kinetic model shown in Table 2.8. However, due to the differences between the model and experiments, it was convenient to express the numerical excitability as the initial acid concentrations. It was found that the numerical excitabilities that better fit the experimental values ε= 0.207, 0.826, and 1.653 M were [H+]0 = 3 M for the lower, [H+]0= 10 M, for the middle, and [H+]0= 15 M for the higher cases respectively. These values were estimated by fitting the experimental values with the simulations using the GNU software COPASI. Figure 5.1 presents a comparison between the numerical and experimental temporal dynamics for three different excitabilities. As can be observed, the model induction time and the amplitude of the oscillatory region decreased inversely to the excitability similarly to the experiments. The absence of oscillations in the low excitability case (ε= 0.207 M) was also observed in the numerical simulations. Also, the last long-period oscillation observed in the experimental cases before the beginning of the saturation region was also reproduced by the model. The oscillatory dynamics of the system was also well reproduced by the model. Figure 5.2 compares the simulations results (Figure 5.1(a)) with the experiments (Fig. 5.2(b)), for a high excitability case [H+]0= 15 M and ε= 1.653 M respectively. In both cases, the oscillatory region is marked inside Chapter 5. Numerical Results domain. ρmand κare the density of the mixture and the permeability respectively (both dependent on the chemical concentrations). For the sake of simplicity, the precipitate was included in the model as a secondary fluid instead of discrete solid particles. From the two-phase flow theory [157, 124], the density of a two-phase mixture can be modeled as: ρm(Ci) = βlρl(Ci)+βsρs(5.3) where βland βsare the volume fractions of the liquid phase and the solid phase respectively and βl+βs = 1. The density of the liquid phase was assumed to vary linearly with respect the chemical species concentrations [197]: ρl(Ci) = ρ0[1+∑ i αiCi](5.4) where, αi=1 ρ0 ∂ρ ∂Ci (5.5) are the solutal expansion coefficients [197] of each reagent (except the [Q·H2Q]) and ρ0is the solvent density. Considering βs=1-βland expressing the liquid volume as Vl=VT–Vs, where VTis the total volume and Vsis the solid volume respectively, the density of the mixture was expressed as a function of the volume fractions as: ρm(Ci) = (VT−Vs VT )ρl(Ci)+(1−VT−Vs VT )ρs(5.6) additionally, Vswas also expressed as a function of the quinhydrone molar concentration as: Vs=γ[Q·H2Q]RMMQ.H2QVT] ρs (5.7) where [Q·H2Q] is the quinhydrone molar concentration, RMMQ.H2Qis the relative molar mass of the quinhydrone and γis an ad-hoc parameter conveniently set to adjust the solid fraction to a suitable value to trigger the fingering instability in a reasonable computational time. For a fixed γ, the qualitative behavior of the experimental system was reproduced by only changing the system excitability and density jump in the same way as in the experiments. Regarding the permeability κ, as explained in Shukla et al [176], the formation of a precipitate affects the permeability of the porous matrix where it is formed. Similar to the cited work, the permeability κ=κ([Q·H2Q]) was defined as: κ([Q·H2Q]) = κ0exp(−Rκ([Q·H2Q] c0 )) (5.8) where κ0is the permeability in absence of precipitate ([Q·H2Q] = 0 M, κ0=a2/12). Defining κm=κ([Q·H2Q]) =c0,Rκcan be calculated as Rκ= ln(κ0/κm) similar to Shukla et al [176]. A positive value of Rκindicates that the precipitate locally reduces the permeability of the porous 90 DAR´ IO MART´ IN ESCALA VODOPIVEC matrix [176]. For the present simulations, Rκ= 0.75. This parameter was estimated ad hoc according to the experimental observations. All the remaining parameters and references used to simulate the RDC system that best fit the experimental conditions and results are summarized in Table 2.10 of Section 2.2. 5.4 Non-Linear RDC Simulations 5.4.1 Descriptive Analysis Figure 5.8 shows a comparative plot between three different perspectives: ferroin concentration, Q·H2Q concentration and Density (Figs. 5.8(a-c) respectively) for the case [H+]0= 15 M (high excitability condition) and [Na2SO4]0= 0.200 M (largest density jump) for five different time steps. The finger onset started after the reaction-diffusion pattern and was naturally produced by the generation of Q·H2Q. Also, the finger shape and its dynamics shown similarities with those observed experimentally. As can be seen, the model showed a very good agreement with the experiments (Fig. 3.1) Figure 5.8: Comparative between (a) Ferroin concentration field, (b) quinhydrone concentration field and (c) density field for a simulated case where [H+]0= 15 M and [Na2SO4]0= 0.200 M which is qualitatively equivalent to the experimental counterpart where ∆ρ= 0.011 g/cm3and ε= 1.653 M. This Figure was taken from Escala et al [61]. 5.4.2 Instability Variation as a Function of ∆ρand ε In Figure 5.9, the density profile evolution is shown for four different simulated scenarios. The simulated cases correspond with the extreme cases presented in Figure 3.2, that is, lowest and highest excitability cases, and smallest and largest density jumps. Two simultaneous snapshots are presented for each case corresponding with the ferroin and Q·H2Q concentration fields, analyzed at the same time. Fingering instability was produced at the interface by a A + B →C mechanism, where the quinhydrone is produced by the accumulation of Q and H2Q. The system initially showed the typical reaction-diffusion 91 Chapter 5. Numerical Results patterns and once the fingering instability was triggered, the chemical waves remained moving around the fingers. The shape of the fingers also shown similitude with those observed experimentally. Figure 5.9: Temporal variation of the density profile for four different cases: (a) lowest excitability, largest density jump ([H+]0= 3 M, ∆ρ= 0.011 g/cm3), (b) highest excitability, largest density jump ([H+]0= 15 M, ∆ρ= 0.011 g/cm3), (c) lowest excitability, lowest density jump ([H+]0= 3 M, ∆ρ= 0.002 g/cm3) and (d) highest excitability, lowest density jump ([H+]0= 15 M, ∆ρ= 0.002 g/cm3). For each case, the two frames represent the concentration fields at the end of each simulation for the species ferroin (upper pic) and quinhydrone (lower pic). The density profiles were obtained by taking the density values across the line indicated in (a). This Figure was taken from Escala et al [61]. For a low excitability condition (Figure 5.9(a)), the formation of quinhydrone was not strong enough as to induce fingering instability. This fact can be appreciated in the density profile, whereas the initial density jump was large, the system became more stable and the small amount of Q·H2Q generated barely changed the density as to locally destabilize the system. This fact was not observed in the Figure 5.9(c), where fingering instability occurred. For this case, as the initial density jump was smaller compared with Figure 5.9(a), the generation of quinhydrone produced a local destabilization of the system, triggering the fingering instability. In Figures 5.9(b) and 5.9(d), as the system was more excitable, the quinhydrone was rapidly generated inducing a faster fingering instability in both cases. For the more stable case (Figure 5.9(b)), the finger onset took longer to appear compared with Figure 5.9(d). These results can be compared with those reported in Figure 3.2 and Figures 3.3-3.4. 5.4.3 Numerical Measuring Observables The numerical measuring observables were calculated identically to the experimental case by measuring the space-time plot obtained from the simulations. The results are presented in Figures 5.10 and 5.11. Alike the experiments, the chemical induction time (tind−C) decreased when the excitability was increased. Similar to the experimental case, this observable did not depend on the initial density jump 92 DAR´ IO MART´ IN ESCALA VODOPIVEC and it was only influenced by the initial chemical conditions. The chemical period (TC) also showed a good agreement with the experiments. This observable decreased when the excitability was increased (Fig. 5.11(a)). Regarding the hydrodynamic induction time (tind−H), a relative increment of this value was observed when the hydrodynamic jump was increased. The changes in the finger morphology also shown similarities with their experimental case. It was possible to see an increment in the finger wavelength for lower excitability values (this was better observed for large density gradients). This effect became less significant for larger excitabilities (Figure 5.11(b)). All these results agreed with the experimental observations presented in Figures 3.3-3.4. Figure 5.10: STP comparison between all the simulated cases presented in Fig 5.9: (a) ([H+]0= 3 M, ∆ρ= 0.011 g/cm3), (b) ([H+]0= 15 M, ∆ρ= 0.011 g/cm3), c) ([H+]0= 3 M, ∆ρ= 0.002 g/cm3) and d) ([H+]0= 15 M, ∆ρ = 0.002 g/cm3). In all cases the chemical and the hydrodynamic induction times are indicated. All the plots were obtained by measuring the y-line indicated in the Fig. 5.8(a). The space and time dimensions are: y= 1 cm, t0= 0 and tf= 220 s respectively. Figure taken from Escala et al [61]. 93 Chapter 5. Numerical Results Figure 5.11: Qualitative comparison of the numerical observables for the cases presented in Figure 5.9. These results were obtained using the same methodology used for Figures 3.3-3.4 (More details are included in Appendix Section C.1). (a) The chemical induction time (tind−C) and the chemical period (TC) were compared for two different excitabilities (given by [H+]0). The differences in period and induction time qualitatively agreed with the experimental results previously presented. (b) the finger wavelength λHwas compared for different situations indicated at the bottom of each plot. Similar to the experiments, large wavelengths were observed for lower excitabilities and vice versa. This phenomenon was less significant in cases where the density gradient was increased, due to the increment in the stability of the system. This fact also agreed with the experimental section. This Figure was taken from Escala et al [61]. 5.5 Chapter Discussion The numerical simulations introduced in the present chapter demonstrated the efficiency of existing models of the BZ-CHD reaction. The models were also capable to predict the spatio-temporal dynamics in a specific experimental context. The viability of the full model to reproduce the main dynamics of the batch system was demonstrated. This was especially appreciated in the chemical characteristic times (tind−Cand TC) with a high degree of reliability. Both observables were well represented by the model in qualitative and quantitative terms. In addition, the model predicted the accumulation of Q and H2Q after the oscillatory region. This was fundamental, not only to confirm the hypothesis made over the occurrence of the saturation region but also to model the quinhydrone complex formation quantitatively. The skeleton model reproduced the main characteristics of the full system as well. However, some differences were observed due to the large simplifications made over the set of equations. Nevertheless, it was shown that the use of this simplified model was suitable to simulate the RD and the RDC systems. The modifications made to both models (by including the Q·H2Q formation), showed a remarkable agreement with the dynamics of the batch system observed through UV-Vis spectroscopy. This was particularly noticeable in the full model, where the quinhydrone formation coincided with the saturation 94 DAR´ IO MART´ IN ESCALA VODOPIVEC region of the spectrum. The development of a convective model and its implementation by non-linear simulations was able to “naturally” reproduce the fingering instability. The convective and oscillating dynamics of the Hele-Shaw cell experiments were, at least qualitatively, perfectly reproduced by the RDC model, showing comparable variations in all the numerical observables. The control simulations showed that the instability only occurred if the chemical model was modified. This demonstrated the major role played by the quinhydrone formation in the experimental observations. 95 Conclusions This part of the thesis presented a detailed description of the chemically induced buoyancy-driven fingering instability. The system was fully characterized by studying the influence of the most relevant parameters of the system, the excitability and the density gradient. The use of these two parameters was fundamental not only to obtain an extensive description of the system but also to unveil the mechanism behind the instability formation. In this sense, the effectiveness of handling the density gradient and the excitability independently was the key to propose a suitable hypothesis regarding the observed phenomena. The major influence of chemistry in the system behavior was strongly demonstrated. The numerical observables quantified in the convective framework, with exception of those associated purely with the hydrodynamics, were conserved in both, the batch experiments and the numerical models. This was fundamental to use such models in the system characterization. In this sense, the instability observed was characterized to show a particular behavior, exhibiting a high degree of synchronization with the chemical dynamic. However, the finger formation per se did not specifically depend on the oscillatory behavior as it could be obtained just with the formation of quinhydrone in absence of oscillations. Nevertheless, it was observed that when both phenomena were present, a bi-directional interaction between both instabilities occurred. This demonstrated that in such a specific condition chemo-hydrodynamic synergies occurred that were produced by similarities in timescales. It was possible to unveil the whole process of finger formation as an oversimplified mechanism, such as the classical A + B →C, where the role of C was played by the non-soluble chemical complex quinhydrone. The detailed characterization of the system made it possible to propose a modification of the existing reaction mechanism that, once coupled with the hydrodynamic part, was able to numerically reproduce the chemically induced density-driven fingering instability. The proposed model, unlike the simple A + B →C case, can be controlled by a broad variety of parameters, producing a much richer ensemble of possible behaviors. It was systematically demonstrated, that a hydrodynamic instability can be triggered and controlled by a chemical reaction at the interface. This phenomenon can be used to understand why typical convective systems in Nature do not behave as theoretically expected in purely hydrodynamic environments, and why the chemical interactions need to be considered. On the other hand, the methods here exposed can be used to design more sophisticated experimental systems where the hydrodynamic instabilities can be controlled by a chemical reaction with countless applications in industry. PART II: VISCOUS FINGERING INSTABILITY DRIVEN BY pH-SHIFTING REACTIONS Appendix A. Preparation of Stock Solutions A.1.5 Sulfuric Acid Solution The sulfuric acid (H2SO4) stock solution was prepared from concentrated (95-98 %, 1.840 g/cm3) H2SO4solution (Sigma-Aldrich, CAS: 7664-93-9, MW: 98.08 g/mol). The stock was obtained by adding gently 356.3 mL of acid into 250 mL of doubly distilled water. As the dissolution process is highly exothermic, the flask must be constantly cooled. Once it was diluted, the volume of the solution was completed to the mark (1000 mL) at room temperature. A.1.6 Sodium Chloride Solution The sodium chloride solution was prepared by diluting 0.584 g of solid salt (Sigma-Aldrich, CAS: 7647-14-5, MW: 58.44 g/mol) into 100 mL of doubly distilled water, obtaining a final concentration of 0.1 M. A.1.7 Reactive Mixture: Solutions 1 and 2 The bubble-free recipe of the Belouzov-Zhabotinsky reaction was separated into two independent and non-reactive solutions. Solution 1 was prepared by mixing 2.91 ml of CHD stock, 0.24 ml of ferroin stock and four different volumes of Na2SO4stock: 1.3 ml, 1.55 ml, 2.05 ml, and 2.3 ml respectively. Solution 2 was prepared by mixing 1.42 ml of NaBrO3 stock, and 0.24 ml of ferroin stock. In both solutions, different volumes of H2SO4stock were added in equal concentration to avoid acid gradients. Specifically: 2 ml, 4 ml, 6 ml, and 8 ml. Finally, doubly distilled water was added to obtain a final volume of 15 ml for each solution. Solution 1 shows a characteristic red coloration as the ferroin is in a reduced state while solution 2 shows a blue coloration due to the oxidized state of the ferroin (ferriin). A.1.8 Protocol to prepare the BZ-Agarose Gels The agarose stock solution was prepared by dissolving 1.5 g of Agarose TYPE I, Low EEO (Sigma, CAS: 9012-36-6) into 100 mL of doubly distilled water obtaining a final concentration of 1.5 wt%. The dissolution was done in boiling water. Once it was dissolved, the mixture was kept warm at not less than 45°C (which is closer to the gelation temperature). For preparing the non-reacting agarose gels for the system presented in Appendix B, two individual solutions were prepared using the same stocks indicated in this section. Gel 1 was prepared by adding 1.46 mL of CHD stock, 0.12 mL of ferroin stock, 2mL of agarose stock and, 3.92 mL of doubly distilled water to obtain a final volume of 7.5 mL. Gel 2 was prepared by adding 0.71 mL of bromate stock, 0.12 mL of ferroin stock, 2 mL of agarose stock, and 4.67 mL of doubly distilled water to obtain a final volume of 7.5 mL. In both cases, the agarose was added at last to avoid premature gelation. Solutions were constantly stirred to facilitate a homogeneous mixture. To filling the Petri dish, a stopper was added to half of the Petri dish (silicone rubber of 0.5 mm of thickness is recommended). Once the agarose was added into Solution 1, the mixture was poured into the other half of the Petri dish. Once it was gelified, the stopper was removed and the second solution was added. Once the gel was obtained, the excitability was changed by adding 2 mL of diluted sulfuric acid on top of the gel (the acid can not be added directly in the solution formulations as it interferes with the gelation process. The agarose only gelifies at neutral or basic pH). The acid was distributed homogeneously. The experiment started once half of the Petri dish switched from red coloration (reduced state) into blue coloration (oxidized state). 202 DAR´ IO MART´ IN ESCALA VODOPIVEC A.2 Stock Solutions Used for Part II A.2.1 Formaldehyde Solution The formaldehyde stock was a commercial concentrated Formalin solution (Sigma-Aldrich, CAS: 50-00-0, MW: 30.3 g/mol). A.2.2 Sodium Sulfite Solution The sulfite stock was prepared from solid NaBrO3anhydrous salt (Sigma-Aldrich, CAS: 7757-83-7, MW: 126.04 g/mol) by diluting 25.21 g of salt into 100 mL of doubly distilled water, obtaining a final stock concentration of 2 M. A.2.3 Poly(Acrylic Acid) [PAA] solutions The PAA solution was prepared from Poly(Acrylic Acid) (Sigma-Aldrich, CAS: 9003-01-4, Average Mv ∼4000000). The stock was obtained by dissolving 1 g of PAA into 180 mL of doubly distilled water at 80°C to facilitate solubility. After solubilization, the mixture was cooled down to 23°C and the final volume was kept at 200 mL obtaining a final concentration of 0.5 wt%. For the control experiments, a short-chain PAA molecule was used (Mv sin 450000). The stock solution was obtained by dissolving 5 g of PAA into a total volume of 100 mL of doubly distilled water obtaining a final concentration of 5 wt%. The dissolution procedure was the same used for the long-chain PAA. A.2.4 Sodium Bisulfite Solution The Bisulfite stock solution was prepared from metabisulfite sodium salt (Na2S2O5(Sigma-Aldrich, CAS: 7681-57-4, MW: 190.11 g/mol), by diluting 19.011 g of salt into 100 mL of doubly distilled water, obtaining a final stock concentration of 1 M. A.2.5 Sodium Hydroxide Solution The sodium hydroxide stock was a commercially available NaOH solution 5.0 M (Sigma-Aldrich, CAS: 1310-73-2, MW: 40.00 g/mol) A.2.6 Gluconolactone Solution The gluconolactone stock solution was freshly prepared for each experiment from reagent grade D-(+)-Gluconic acid δ-lactone (Sigma-Aldrich, CAS: 90-80-2, MW: 178.14 g/mol). The stock was obtained by diluting 0.356 g of gluconolactone in 10 ml of doubly distilled water. This solution has been used always fresh and in less than 300 seconds after being prepared to avoid hydrolysis effects. A.2.7 Bromothymol Blue Indicator The color indicator is a 0.4 wt% hydroalcoholic solution of Bromothymol blue prepared by dissolving 1 g of Bromothymol blue sodium salt powder (Sigma) into 50 ml of a 96% ethanol solution diluting up to a final volume of 250 mL by adding 200 mL of doubly distilled water. The C.I. shows a yellow color for pH values below 6 (acidic state), green color for pH between 6-7 (neutral state), and blue color for pH values above 7 (basic state). 203 Appendix A. Preparation of Stock Solutions A.2.8 Displacing Solution Mixture Preparation As shown in Part II, there is not a unique recipe for the displacing solution because the sulfite concentration was varied. However, a base formulation was used to study the effects of the formaldehyde in the instability development. This base formulation was prepared by mixing 5 mL of PAA stock, 0.195 mL of sulfite stock, 0.300 mL of C.I. stock, and 0.205 mL of doubly distilled water. This solution shows a green coloration and its pH is between 6.7-7. A.3 Stock Solutions Used for Part III A.3.1 Formaldehyde Solution Same recipe as used for Part II. A.3.2 Sodium Sulfite Solution Same recipe as used for Part II. A.3.3 Poly(Acrylic Acid) [PAA] solution Same recipe as used to prepare PAA Mv ∼4000000 for Part II. A.3.4 Sodium Carbonate Solution The sodium carbonate (Na2CO3) solution used for the control experiment 5 (C5) was prepared from sodium carbonate anhydrous powder (Sigma-Aldrich, CAS: 497-19-8, MW: 105.99 g/mol), by diluting 21.19 g of powder into 100 mL of doubly distilled water obtaining a final stock concentration of 2 M. A.3.5 Bromothymol Blue Indicator Same recipe as used for Part II. A.3.6 Gluconic Acid Solution (Solution B) Solution B is a concentrated aqueous solution of gluconic acid 2.0 mol/kg (∼ =1.66 M), This solution was obtained by diluting 17.81 g of D-(+)-Gluconic acid δ-lactone (Sigma-Aldrich, CAS: 90-80-2, MW: 178.14 g/mol) into 50 g of doubly distilled water. The solution was left to rest a complete day to ensure the full conversion of the gluconolactone into gluconic acid by hydrolysis. A.3.7 Solution A Mixture Preparation Solution A was prepared by mixing 5 mL PAA stock, 0.195 mL of SO32 – stock, 0.300 mL of C.I. stock, 0.150 mL of Formaldehyde stock, and 0.055 mL of doubly distilled water. This solution shows a blue coloration and its pH is between 11.5-12. 204 Appendix B Reaction-Diffusion Systems Abstract:This Appendix will introduce the main results obtained from less complex reaction-diffusion (RD) systems. These results were used as guidance to develop and understand the most complex scenario presented in Part I. In this sense, the first observation and the analysis of the dynamic of a spatially extended BZ-CHD oscillator were done in a 1D capillary system. This study aimed to obtain relevant information about how the typical oscillatory behavior of a homogeneous system may be affected by separating part of the reagents into two different non-reactive solutions. With those results, the next step was to extend the setup of the capillary system into a 2D non-convective experimental setup and to study the influence of a spatially extended configuration in the pattern formation and thus, to better understand the role played by the chemistry in the convective system. In the further sections, an extensive description of those previous non-convective systems is presented, showing both experimental and numerical methods and results. B.1 Materials and Methods B.1.1 Capillary System 1D experiments were carried out in the capillary system schematized in Figure B.1. The cylindrical capillary reactor was built in borosilicate with an inner diameter of 0.05 mm ± 0.01 mm, 2 ± 0.1 mm outer diameter, and 70 ± 1 mm length. The small inner diameter avoids any convection or 2D reaction-diffusion pattern, ensuring a purely 1D dynamic. To obtain the interfacial initial condition, the following method was used: Firstly, Solution 1 was introduced into the capillary by using capillary forces. Once half-filled, Solution 2 was introduced from the same opening and in the same way until the reactor was filled. The experiments were recorded using a CMOS camera (PixeLink PL-B776U) connected to a computer with a total experimental time of 6 hours. The species of each solution are indicated in the schematic of Figure B.1. Chemical Recipe The recipe used is the same presented in Table 2.1. As these experiments were also part of a set of control experiments, just a few experiments were performed covering low, medium, and high excitability cases. More specifically, those corresponding to ε=0.207 M, ε=0.826 M, and ε=1.653 M. As described in the previous section, the reactor has an inner diameter of 0.05 mm which makes it difficult to observe the liquids inside the capillary even if there are colored. For such a reason, the experiments were done increasing the ferroin concentration up to four times the concentration listed in Appendix B. Reaction-Diffusion Systems Figure B.1: Capillary reactor used in 1D reaction-diffusion experiments. The capillary was built in borosilicate. Both solutions were introduced into the system using capillary forces. Table 2.1. Even though the increment in ferroin could affect the system temporal dynamic, it did not influence the qualitative behavior of the results. B.1.2 Non-Convective Agarose-Based System A pure 2D reaction-diffusion system was performed in a Petri dish as presented in Figure B.2. The Petri dish was chosen for its simplicity and reliability. To obtain a non-convective system, the BZ-CHD reaction was mixed with an agarose solution. The experiments were carried out in a 7 cm diameter Petri dish. The agarose solution was preheated to avoid a fast gelification. The initial condition observed in Figure B.2(a) was obtained by putting Solution 1 until half of the Petri dish. Solution 2 was added after gelation of Solution 1. Both gels were not reactive as the sulfuric acid was not present in the recipes (agarose does not gelify at low pH conditions). However, once the initial gel configuration was obtained, the H2SO4was added into the system from the top and distributed homogeneously through the entire the gel surface. As the acid diffuses through the gel, the ferroin present in Solution 2 oxidizes into ferriin, obtaining a similar system presented in Figure B.2(b). Analogously to the 1D experiments, the 2D system was recorded using a CMOS camera (PixeLink PL-B776U) for a total experimental time of 6 hours. Similar to the 1D case, this set of experiments were used to analyze the pattern dynamic at the interface and evaluate the influence of the chemistry in the fingering instability observed in the full convective system. All the experiments realized under this configuration were made without replicas as they were used as well as a control system. In this sense, the same conditions of high, middle, and low excitability were studied. Chemical Recipe In this case, the original recipe was slightly modified to make the system non-convective. Thus, 2 mL of agarose 1.5 wt% was added to each solution to make a gelified version of the convective system. The recipe is presented in Table B.1. The system excitability was varied in the same manner as the capillary experiments being ε=0.207 M, ε=0.826 M, and ε=1.653 M the studied cases. The excitability was varied by adding different acid concentrations over the Petri dish after gelification. The reaction started once the Ferroin on the gel 2 became Ferriin. 206 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure B.2: (a) Schematics of the 2D reaction-diffusion system built in a Petri dish. (b) Image of an experimental Petri dish after the ferroin conversion into ferriin. In this experiment, both solutions were mixed with agarose 1.5 wt% to produce a pure 2D reaction-diffusion system. The reaction starts once the acid is homogeneously poured into the Petri dish from the upper side. Recipe for Gel 1 Recipe for Gel 2 Species Concentration (M) Species Concentraion (M) CHD 0.291 BrO3–0.142 [Fe(phen)3]2+ 0.4x10−3[Fe(phen)3]3+ 0.4x10−3 S1 volume of Agarose 1.5 wt%: 2 mL S2 volume of Agarose 1.5 wt%: 2 mL Gel 1 volume: 7.5 mL Gel 2 volume: 7.5 mL Table B.1: Recipe used in the non-convective agarose 2D-Reaction Diffusion System. B.2 Numerical Models B.2.1 Governing equations Both reaction-diffusion systems were modeled using the following general set of partial differential equations: ∂Ci ∂t=DCi∇2Ci+Ri(Ci)(B.1) where Riand Ciare the net reaction rates and the concentration of the involved species respectively. The net reaction rates were obtained from the skeleton model presented in Table 2.9 including the quinhydrone formation equation. Analogously to the convective simulations, the modified skeleton model was used in place of the 207 Appendix B. Reaction-Diffusion Systems full model. This was necessary due to the large number of memory and resources needed that make its use prohibitive. This model was far enough to represent the qualitative behavior of the diffusive systems. All simulations were done using a Finite Volume Method solver implemented in the commercial CFD software Ansys Fluent® version 19.2 [11, 12]. All results were obtained by using the SIMPLE solver coupled with a stiff-chemistry solver. The time step was automatically controlled by the software and both space and time, were discretized using the first-order upwind method. The total computational time was set to tf= 3.5 min (real-time). The difference between the experimental and computational final times is due to the simplifications of the modified skeleton model used to simulate the spatially extended systems. B.2.2 1D-RD System Model Setup The capillary system was simulated using a 1.5D approach. The numerical domain (D) consisted of a rectangle of width Land arbitrary height, discretized by using a structured mesh of 181 elements in the x-direction and 1 element in the y-direction as shown in Figure B.3. The use of a 1.5D domain was preferred to facilitate the comparison between experiments and simulations. Zero diffusive flux conditions were set as boundary conditions. Figure B.3: 1.5D numerical domain used to simulate the capillary system. The mesh consists of 181 elements in the x-direction and 1 element in the y-direction. This type of domain provides a dynamic color profile of the species concentration similar to the experimental counterpart without increasing the computational cost. The species spatial configuration as presented in Figure B.3 is given by the following piecewise function: Cai(x,0) = ([Ca]0(x)(ξr(x)+1)x<L/2 0x⩾L/2 Cbi(x,0) = (0x<L/2 [Cb]0(x)(ξr(x)+1)x⩾L/2 [H2SO4]0(x,0) = k,∀x∈D (B.2) where [Ca]0are: [CHD]0and [Fe(phen)3]2+ 0.[Cb]0are: [NaBrO3]0and [Fe(phen)3]3+ 0.r(x,y)is a normally distributed random function with amplitude ξ= 10−2. The [H2SO4]0was set constant in all the domain. B.2.3 2D-RD System Model Setup For the 2D system, the numerical domain (D) consisted of a squared mapped mesh of area L2 discretized by 181 elements in both directions, xand y. A schematics of the domain and the spatial configuration of the reactant species are presented in Figure B.4. The initial conditions were set using the same expression used for the RDC simulations (Eq. 2.2.3), except for [Na2SO4]0, which was excluded from the RD system as the density was not taken into account. All boundaries conditions were set as zero diffusive flux. 208 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure B.4: 2D numerical domain used to simulate the RD model. The mesh consists of 181 elements in the horizontal direction and 181 elements in the vertical direction. B.3 Experimental Results B.3.1 1D Capillary System The dynamics of the capillary system were observed through space-time plots (hereafter STP). These plots were obtained by measuring the pixel changes across a reference line located in the fluid reservoir (line indicated as xin Fig. B.1(a)). The three studied excitabilities are compared in Figure B.5. As can be seen, the overall behavior of the system agrees with the convective results. This was verified by studying the chemical observables. Both the induction time (tind −C) and chemical period (TC) decreased with the increment of excitability. In all cases, the wavelength also increased in time. This was particularly noticeable just before the oscillation region ended. These behaviors were expected as this system was only driven by chemistry. The generation of the quinhydrone complex can be appreciated as a strong reddish mark for the mid and high excitability cases. B.3.2 Non-Convective Agarose-Based System Figure B.6 shows the effect of changing the excitability in the agarose based non-convective system. Snapshots were taken directly from the experimental observations. As can be seen, for the lower excitability (ε= 0.207 M, Fig. B.6(c)), the system did not show any significant behavior in the experimental time. Only the diffusive displacement of the interface was observed. This indicates that such a chemical condition was not enough to develop spatial structures in the established experimental time (t= 6h). For the middle excitability (ε= 0.826 M, Fig. B.6(b)), traveling waves were observed moving through the interface, additionally to the initial front displacement. Finally, in the higher excitability case (ε= 1.653 M, Fig. B.6(a)), waves were observed to occur faster than the previous case. This also agreed with the convective case, where the increment of the excitability decreased the induction time and the pattern wavelength. The effect of the excitability on the system behavior is better observed in the space-time plots obtained from the 2D system. Plots are presented in Figure B.7 and were obtained by measuring in time the spatial region indicated with yin Figure B.6(a). As can be seen, both the delay in the induction time 209 Appendix B. Reaction-Diffusion Systems Figure B.5: Space-Time plots obtained from the 1D reaction-diffusion capillary system for high (ε= 1.653 M), (b) middle (ε= 0.826 M), and (c) low (ε= 0.207 M). Figures were constructed by taking a profile line across the longitudinal axis of the capillary. In all cases: [CHD]0= 0.291 M, [BrO3–]0= 0.142 M, [Fe(phen)32+]0= [Fe(phen)33+]0= 0.4x10−3M. and the front displacement were visible for middle and higher excitabilites. Unlike the 1D experiments, this system exhibits a much larger activity showing interface oscillations for a long time. 210 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure B.6: 2D reaction-diffusion experiments for three different excitability conditions (a) ε= 1.653 M, (b) ε = 0.826 M, and (c) ε= 0.207 M. Patterns similar to the ones observed in the convective case were observed for the middle and higher excitability cases. The remaining initial concentrations were fixed as: [CHD]0= 0.291 M, [BrO3–]0= 0.142 M, [Fe(phen)32+]0= [Fe(phen)33+]0= 0.4x10−3M. Figure B.7: STPs obtained from the non-convective agarose-based system, The changes in wavelength, induction time and front displacement are clearly appreciated. In all cases, the remaining initial concentrations were set as: [CHD]0= 0.291 M, [BrO3–]0= 0.142 M [Fe(phen)32+]0= [Fe(phen)33+]0= 0.4x10−3M. 211 Appendix C. Image Analysis Techniques Figure C.1: Method used to calculate the chemical experimental observables (TCand tind−C) via space-time-plots (STP). These results were used in Figure 3.3. (a) Specifies the methodology used to calculate by indicating the spatial locations used to generate the STP (marked as Yin the pic). In (b), the dashed lines indicate the region of the STP used to calculate the plots observed in (c). tind−Cwas measured directly from the STPs, while TC was calculated using the line profiles presented in (c). Each observable was recorded in the same region of time. Particularly, TCwas measured once oscillations started. t0= 0 h and tf= 6 h. Figure C.2: Method used to calculate the hydrodynamic experimental observables (λHand tind−H) via space-time-plots (STP). These results were used in Figure 3.4. (a) Specifies how the STP was calculated. x indicates the spatial location used to generate the STP. In (b), the dashed lines indicate the region of the STP used to calculate the plots observed in (c). tind−Hwas measured directly from the STP, λHwas calculated using the line profiles presented in column (c). Each observable is recorded in the same region of time. t0= 0 h and tf= 6 h. C.2 Analysis Methods Used for Part II Traditionally viscous fingering patterns have been studied using morphological description tools like the fractal dimension and denisty area. The election of a different methodology lies in the type of pattern given by the experiments. In this work, due to the specific aspect of the experimental instabilities, the circularity was chosen as the main tool to describe the evolution of the fingering pattern. However, to check the feasibility of this method, results were also compared with the density area tool. C.2.1 Circularity Calculation The circularity was used to obtain the quantitative measurements of both numerical and experimental results. The algorithm calculates the circularity of the shape of a binary mask extracted from the fingering pattern formed by the displacing solution once the instability is fully developed. The mask extraction was performed by using an adaptive selection tool (magic wand) and was then converted into a binary mask by applying a threshold function. As all the experiments were done using a calibration bar, the relation between pixels and real spatial units is known. The circularity calculation was later done automatically by calculating the perimeter and the area from the experimental/numerical 218 DAR´ IO MART´ IN ESCALA VODOPIVEC fingering patterns. A schematic of this procedure is presented in Figure C.3. All circularity measurements were done on the final frame of each experiment. Figure C.3: Schematic of the procedure used to obtain a circularity value from the experimental images. The circularity value was calculated from the binary mask obtained extracted from the experimental/numerical figure. C.2.2 Density Area Calculation The density area dAis defined as the ratio between the area occupied by the fingering pattern (AreaP) and the radius of the longest finger RMax and is calculated from the following expression [65, 142, 139]: dA=AreaP πR2 Max (C.1) As previously commented, the results obtained using the circularity were compared with those obtained calculating dA. Figure C.4(a) shows the dAcalculated for Q = 20 mL/min. These results must be compared with the results of Figure 8.6. As can be noted, this parameter recovered the same main features as the circularity. Because the patterns were not very elongated, the changes in density area were restricted to a small fraction of the radius and, thus, the actual uncertainties became more important. This is the reason behind the use of the circularity presented in the results of Part II. This demonstrated the consistency of the results independently on the parameter chosen to describe them. 219 Appendix C. Image Analysis Techniques Figure C.4: (a) Finger density area dAcalculated for Q = 20 mL/min on Figure 8.6. Both results agree with those obtained using the circularity. (b) dAcalculated at fixed distance d = 108 mm as a function of the Formaldehyde]0. C.3 Analysis Methods Used for Part III C.3.1 Time-Dependent Circularity Calculation The procedure to calculate the time-dependent circularity is presented in Figure C.5. In this case, every analyzed frame of each studied case was processed by the Hematoxylin and Eosin (H&E) color deconvolution algorithm (similar results may be obtained by using other algorithms included in the software, however, the default algorithm was the simplest choice). Once processed, three-color components were obtained. A binary mask was then created from one of the color components. For Case I, the best results are obtained binarizing the first color component (Color 1 in the schematics of Fig. C.5). For Case II, the second color component was chosen (Color 2 in the schematics of Fig. C.5). Once the binary images were obtained, the software calculated the circularity following the definition. As several frames of a complete experimental run were processed, the evolution of the circularity can be plotted as a function of time. C.3.2 Average Displacing Profile The methodology to obtain the average displacing interface for the direct experiments shares some similarities with the previous procedure and it is presented in Figure C.6. Several space-time plots (STP, indicated as P1,P2,...,Pn) were obtained by taking radial slices of a complete experimental run. These STPs were processed by using the same H&E color deconvolution algorithm used to measure the circularity (Fig. C.5). Three color components were obtained for each set of STPs. Color component 1 was then selected and binarized (this component showed the best results for the profile characterization). The obtained set of binary STPs was then processed by edge detection and binary skeletonization algorithms that converted the image data into a set of numerical curves. From each set of numerical profiles, an average displacement and its dispersion (standard deviation) were calculated. 220 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure C.5: Methodology to calculate the time-dependent circularity for Cases I and II. The frames of every experimental case were processed in FIJI by using a color deconvolution algorithm. From such a process, three-color components were obtained for each frame. Color I and II were used to obtain a binary mask for each type of experiment. From such a mask, the circularity was calculated as indicated in the scheme. A circularity value was obtained for every processed frame. Those values were then represented in a relative time scale for comparison. 221 Appendix C. Image Analysis Techniques Figure C.6: Procedure to calculate the average displacing profile for Case I. In this case, a stack of n-space-time plots was obtained from an experiment by performing a radial reslice operation as indicated in the figure. Every frame of such stack was then filtered by a color deconvolution algorithm obtaining three stacks, one for each color component. The binary mask obtained from color 1 was then processed by an edge detection algorithm obtaining a well-defined profile function. A skeletonize algorithm was finally used to retrieve the space and time coordinates used for statistical calculations. 222 Appendix D Supplementary Results D.1 Results for Part I D.1.1 Solutal Expansion Coefficient Calculation The solutal expansion coefficients (αi) represent the change in density produced by the increment in the concentration of a chemical species [197]. In formal terms, these coefficients are defined as: αi=1 ρ0 ∂ρ ∂Ci (D.1) where ρ0is the density of the pure solvent. These values were used for simulating the RDC system in Part I as indicated in Section 5.3. The solutal expansion coefficients were calculated experimentally by measuring the density variation due to the changes in the concentrations of NaBrO3, Na2SO4, [Fe(phen)3]2+/3+ 0, and CHD at a constant temperature. The ferriin coefficient was assumed to be the same as the ferroin one. The H+coefficient was set ad hoc. The values were obtained from the fitting slope of the experimental measurements. The data, the fitting values, and the coefficient of determination (R2) are shown in Figure D.1. D.1.2 RDC Model Permeability Variation As it was explained in the previous section, the precipitate interacts with the porous matrix by reducing locally the permeability [176]. This implies a positive Rκfactor [176]. Figure D.2 shows the permeability drop generated due to the precipitation formation. The obtained results were similar to those presented in previous works [176]. D.1.3 RDC Control Simulation Figure D.3 shows a RDC control simulation where Equation R1 (quinhydrone formation) was not included in the skeleton model. This simulation is intended to demonstrate that it is not possible to obtain a fingering instability without modifying the original kinetic model. As can be seen, the instability was not observed and its behavior was similar to the RD simulations presented in Appendix B. Appendix D. Supplementary Results Figure D.1: Fitting for the solutal expansion coefficients obtained from experimental data for (a) NaBrO3, (b) Na2SO4, (c) [Fe(phen)3]2+/3+ and (d) CHD. Figure D.2: Permeability as a function of the precipitate concentration for [H+]0= 15 M and ∆ρ= 0.011 g/cm3. At t= 0 s, the permeability inside the numerical domain was constant. Once the precipitate started forming, the permeability got reduced at its vicinity. The plot was obtained by measuring the permeability across a line aligned to the y-direction in the RDC simulations. This Figure was taken from Escala et al [61]. 224 DAR´ IO MART´ IN ESCALA VODOPIVEC Figure D.3: 2D simulations of the skeleton model plus the equations describing the convection in the system for [H+]0= 15 M, ∆ρ= 0.002 g/cm3. As the model does not include the generation of quinhydrone, it is unable to reproduce the fingering instability and a planar interface remains in the system. This Figure was taken from Escala et al [61] 225 Appendix D. Supplementary Results D.2 Results for Part II D.2.1 Effect of the Color Indicator Even though the color indicator facilitates the experimental observation and the image processing, it was necessary to verify if it had any influence on the presented results. In that sense, the effect on the rheology of the system due to the color indicator was measured. In Figure D.4, a comparison between two different samples used in the main experiments is presented. On the one hand, the sample indicated as “With C.I.” corresponds to a solution containing 0.438 wt% of PAA, 0.068 M of SO32 – , and 0.021 wt% of C.I. On the other hand, the curve indicated as “Without C.I.”, corresponds to a solution in which the color indicator was replaced by doubly distilled water. The remaining reagents were kept equal as the colored case. Both curves were compared in a viscosity-shear rate plot measured by the rheology equipment described in Section 6.1.2. As can be seen, no significant change was observed in the viscosity of the solution due to the addition of the color indicator. This demonstrates that in the used concentrations this compound did not affect the rheology of the system. Figure D.4: Study of the effect produced by the color indicator in the overall viscosity of the displacing solution. The black dotted curve corresponds to the viscosity of the displacing solution used in the colored experiment, while the red squared curve corresponds to the viscosity of the solution used in the Schlieren experiments (Fig. 8.8). This demonstrates that the rheology of the system was not affected by the addition of the color indicator. This Figure was adapted from Escala et al [58]. D.2.2 Elasticity Effects and Shear Rate Estimation Non-Newtonian fluids, like the polymeric solutions used in this work, show many interesting characteristics. One of them is the dependence of the viscosity with the shear rate. In particular, the PAA shows a shear-thinning Newtonian behavior. This means that the viscosity of a PAA aqueous solution decreases by increasing the shear rate. Another important characteristic is the elasticity. The elastic property of a polymeric solution is also related with to overlap concentration (c∗), which was introduced in Sections 1.9 and 7.6. The study of the elasticity of the PAA solutions is fundamental to discard any undesired artifact related to it and 226 DAR´ IO MART´ IN ESCALA VODOPIVEC to ensure that all results observed were driven just by the changes in viscosity produced by the studied chemical reactions [139, 140]. In similar works, the elasticity of similar PAA solutions was studied in a simplified manner by measuring the first normal stress difference (N1) [139]. Normal Stresses are caused by shear forces and are typically observed in polymer solutions [70, 211, 215]. Such stresses may appear not only in rheological measurements (especially when liquids are confined in a cone-plate geometry as the one used in Chapter 7) but also inside a Hele-Shaw cell. The magnitude of the normal stress depends not only on the type of fluid but also on the shear rate to which that fluid is exposed. For this reason, it is fundamental to study the elasticity of the PAA solutions in the range of shear rates used. For all results presented in Chapter 7, the shear rate was fixed to a value of 500 s−1. However, inside the Hele-Shaw cell, the shear rate is not constant and had to be measured. Based on the previous work of Nagatsu et al [140, 142, 139], the shear rate inside a radial Hele-Shaw cell can be estimated as: ˙ γf=Q πria2(D.2) where ri= 6.18 cm is the radius of the initial condition obtained from the experiments, and a= 0.25 mm is the gap between both plates. By considering the extreme cases (Q= 2.5 - 10 mL/min), the estimated range of shear rate at the vicinity of the fingertip was: 3.44 ≤˙ γf[s−1]≤27.48 (D.3) The measurement of N1was done by using the same rheological equipment presented in Section 6.1.2. The most concentrated solutions (and expected to be the most elastic ones) shown in Figures 7.3(b) and 7.8(a), were compared with the control solution of PAA and NaOH. The results are presented in Figure D.5. As can be seen in Figure D.5(a)), the control solution (green diamonds curve) exhibits a strong increase in normal stress for ˙ γf>100 s−1. This was expected since a 4x106gmol−10.47 wt% PAA solution gelifies for a broad range of NaOH concentrations due to the extension of the polymer chain [142]. As was discussed in Section 7.6, the PAA dissociation increased the radius of gyration Rgand in consequence, an overlapping scenario was reached. However, the blue dotted curve shows that there was no measurable normal stress found in the range of shear rates used for both the Hele-Shaw experiments and stirred system (500 s−1). 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