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. Da ío Ma ín Escala Vodopi ec
P esen o mi esis, siguiendo el p ocedimien o adecuado al Reglamen o, y decla o que:
1) La esis aba ca los esul ados de la elabo ación de mi abajo.
2) En su caso, en la esis se hace e e encia a las colabo aciones que u o es e abajo.
3) La esis es la e sión de ini i a p esen ada pa a su de ensa y coincide con la e sión en iada en
o ma o elec ónico.
4) Con i mo que la esis no incu e en ningún ipo de plagio de o os au o es ni de abajos
p esen ados po mí pa a la ob ención de o os í ulos.
En San iago de Compos ela, 29 de diciemb e de 2020
Fdo. Da ío Ma ín Escala Vodopi ec
AUTORIZACIÓN
DEL
DIRECTOR
/
TUTOR DE LA
TESIS
HYDRODYNAMIC INSTABILITIES COUPLED WITH COMPLEX CHEMICAL
REACTIONS: CONTROL, CHARACTERIZATION, AND THEIR MODELING
D. Albe o Pé ez Muñuzu i
D. Jo ge Ca ballido-Landei a
INFORMAN:
Que la p esen e esis, co esponde con el abajo ealizado po D. Da ío Ma ín Escala Vodopi ec,
bajo mi di ección, y a
u o izo
su
p esen ación
, conside ando
que eúne l os
equisi os
exigidos en
el R
eglamen o
de Es udios de
Doc o ado de la USC,
y
que
como di ec o de és a
no incu e
en las causas de
abs ención es ablecidas
en Ley
40/2015.
En San iago de Compos ela, 29 de diciemb e de 2020
Fdo. Albe o Pé ez Muñuzu i
Fdo. Jo ge Ca ballido-Landei a

As´
ı queda on e minados los cielos y la ie a, y odo lo que
hay en ellos. Al llega el s´
ep imo d´
ıa, Dios descans´
o po que
hab´
ıa e minado la ob a que hab´
ıa emp endido.
G´
enesis 2:1-2
...Y en el oc a o d´
ıa dio comienzo una esis... que hoy,
a o unadamen e, ha llegado a su in...
Thus he hea ens and he ea h we e inished,
and all he hos o hem. And on he se en h day
God inished his wo k ha he had done, and he
es ed on he se en h day om all his wo k ha
he had done.
Genesis 2:1-2
...And on he eigh h day a hesis began... which
o una ely oday has come o an end...
Queda on as´
ı ema ados o ceo e a e a e odos os elemen os.
Deus conclu´
ıu no d´
ıa sex o a ob a que emp ende a, e o
s´
e imo d´
ıa epousou de odo o seu aballo.
X´
enese 2:1-2
...E no oi a o d´
ıa deu comezo unha ese... que hoxe,
a o unadamen e, chegou ao seu in...
Dedicado a
Mi Familia, Amigos,
Recue dos y C eencias.

Con en s
Summa y ix
Resumen x
Resumo xxi
Nomencla u e and Abb e a ions xx ii
Lis o Figu es xxix
Lis o Tables xxx
Lis o publica ions xxx ii
1 In oduc ion 1
1.1 Homogeneous Sys ems and Chemical Reac ions . . . . . . . . . . . . . . . . . 1
1.2 Typeso Reac ions ................................ 3
1.3 Typeso Reac o s................................. 6
1.4 The Belouso -Zhabo insky Reac ion . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Reduced Kine ic Models o he BZ Reac ion . . . . . . . . . . . . . . 8
1.4.2 The 1,4-Cyclohexanedione-B oma e-Acid Oscilla o y Reac ion . . . . 10
1.5 pH-Shi ingReac ions .............................. 12
1.5.1 Ino ganic pH-Oscilla o s . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5.2 O ganic pH-Shi ing Reac ions . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Reac ion-Di usion-Con ec ion (RDC) Sys ems . . . . . . . . . . . . . . . . . 16
1.6.1 Reac ion-Di usion (RD) Sys ems . . . . . . . . . . . . . . . . . . . . 17
1.7 Hyd odynamic Ins abili ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.7.1 Flow In Po ous Media - Da yc’s Law . . . . . . . . . . . . . . . . . . 21
1.7.2 Hele-ShawCells ............................. 24
1.8 Finge ing Ins abili ies in Hele-Shaw Cells . . . . . . . . . . . . . . . . . . . . 26
1.8.1 Densi y Finge ing Ins abili y . . . . . . . . . . . . . . . . . . . . . . . 26
1.8.2 Viscous Finge ing Ins abili y . . . . . . . . . . . . . . . . . . . . . . . 29
1.8.3 Chemically D i en Finge ing Ins abili ies . . . . . . . . . . . . . . . . 31
1.9 ThePoly(Ac ylicAcid).............................. 34
1.9.1 S uc u e and pH-dependence . . . . . . . . . . . . . . . . . . . . . . 35
1.9.2 Chain Leng h and Concen a ion Regimes . . . . . . . . . . . . . . . . 36
1.9.3 Ionic S eng h E ec s . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Con en s
1.10 The Shadowg aph and Schlie en Op ical Techniques . . . . . . . . . . . . . . 38
1.10.1 Gene ali ies................................ 39
1.10.2 Shadowg aph Technique . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.10.3 Schlie en Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Pa I: Densi y Finge ing Ins abili y D i en by he BZ-CHD Oscilla o 45
Pa I Mo i a ion 45
2 Expe imen al and Nume ical Me hods 47
2.1 Expe imen alMe hods .............................. 47
2.1.1 Hele-Shaw Cell Cons uc ion . . . . . . . . . . . . . . . . . . . . . . 47
2.1.2 Injec ionP o ocol............................. 47
2.1.3 Op ical A angemen . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.1.4 Chemical Recipes and Expe imen al Designs . . . . . . . . . . . . . . 48
2.1.5 Spec oscopy Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.1.6 P ecipi a e Ex ac ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 Nume icalMe hods................................ 55
2.2.1 The BZ-CHD Reac ion Chemical Models . . . . . . . . . . . . . . . . 55
2.2.2 Ba ch Sys em Simula ion . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.2.3 2D Reac ion-Di usion-Con ec ion Model . . . . . . . . . . . . . . . . 57
3 Expe imen al Resul s 61
3.1 Gene al Sys em O e iew . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Desc ip i e Analysis o he E ec o ∆ρand ε.................. 62
3.2.1 Measu ing Obse ables . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Coupled ∆ρand εVa ia ion by Changing [B O3–]0............... 65
3.3.1 Measu ing Obse ables . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 Reac ion-Di usion-Con ec ion In e play . . . . . . . . . . . . . . . . . . . . . 69
3.5 E ec o Va ying [CHD]0............................. 70
3.6 Chap e Discussion ................................ 70
4 De ailed Chemical Analysis 73
4.1 Hele-Shaw Cell Con ol Expe imen s . . . . . . . . . . . . . . . . . . . . . . 73
4.1.1 Ca alys Inhibi ion by NaCl . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.2 Expe imen s Wi hou CHD . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 UV-VisSpec oscopy............................... 75
4.3 P ecipi a eFo ma ion............................... 77
4.4 Nuclea Magne ic Resonance (NMR) Spec oscopy . . . . . . . . . . . . . . . 80
4.5 Chap e Discussion ................................ 82
5 Nume ical Resul s 83
5.1 Equi alence be ween Expe imen s and Reac ion Models . . . . . . . . . . . . 83
5.1.1 Quali a i e Compa ison . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1.2 Quan i a i e Compa ison . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.3 P ecipi a e Fo ma ion P edic ion . . . . . . . . . . . . . . . . . . . . . 86
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5.1.4 Equi alence be ween Full and Skele on Models in Ba ch Sys em . . . . 86
5.2 ModelModi ica ion................................ 88
5.3 Reac ion-Di usion-Con ec ion (RDC) Nume ical Model . . . . . . . . . . . . 89
5.4 Non-Linea RDC Simula ions . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4.1 Desc ip i e Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4.2 Ins abili y Va ia ion as a Func ion o ∆ρand ε............. 91
5.4.3 Nume ical Measu ing Obse ables . . . . . . . . . . . . . . . . . . . . 92
5.5 Chap e Discussion ................................ 94
Pa I Conclusions 97
Pa II - Viscous Finge ing Ins abili y D i en by pH-Shi ing Reac ions101
Pa II Mo i a ion 101
6 Expe imen al and Nume ical Me hods 103
6.1 Expe imen alMe hods .............................. 103
6.1.1 Chemical Recipes and Expe imen al Designs o Dynamic Measu emen s103
6.1.2 Rheological and pH Measu emen s . . . . . . . . . . . . . . . . . . . 105
6.1.3 Radial Hele-Shaw Cell Expe imen s . . . . . . . . . . . . . . . . . . . 105
6.1.4 Ins abili y Se up and P o ocol . . . . . . . . . . . . . . . . . . . . . . 106
6.1.5 Ci cula i y Calcula ion . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Nume icalMe hods................................ 108
6.2.1 The FS-PAA Reac ion Model . . . . . . . . . . . . . . . . . . . . . . 108
6.2.2 The FSG-PAA Reac ion Model . . . . . . . . . . . . . . . . . . . . . 110
6.2.3 Ba ch Sys em Simula ions . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2.4 Reac ion-Di usion (RD) Model Simula ions . . . . . . . . . . . . . . 111
6.2.5 Di usion-Con ec ion (DC) Model Simula ions . . . . . . . . . . . . . 113
6.2.6 Mesh Independence S udy Fo The DC Model . . . . . . . . . . . . . 114
7 pH - Viscosi y Coupling 115
7.1 CouplingMechanism............................... 115
7.2 Sys emDynamics................................. 116
7.3 FS-PAA Sys em Cha ac e iza ion . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3.1 [PAA]0Va ia ion............................. 117
7.3.2 [SO32– ]0Va ia ion............................ 118
7.3.3 [Fo maldehyde]0Va ia ion........................ 119
7.4 FSG-PAA Sys em Cha ac e iza ion . . . . . . . . . . . . . . . . . . . . . . . . 119
7.4.1 [GLN]0Va ia ion............................. 119
7.5 The FS-PAA Al e na i es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.5.1 The E ec o Using HSO32– andNaOH................. 121
7.5.2 The E ec o Using a Sho -Chain PAA Molecule . . . . . . . . . . . . 122
7.6 O e lapConcen a ion .............................. 122
7.7 Chap e Discussion ................................ 123
iii
Con en s
8 Viscous Finge ing Induced by he FS-PAA Reac ion 125
8.1 Gene al Sys em O e iew . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.2 Desc ip i eAnalysis ............................... 126
8.2.1 FlowRa eE ec ............................. 127
8.2.2 [Fo maldehyde]0Va ia ion in he Displaced Solu ion . . . . . . . . . . 127
8.2.3 [SO32– ]0Va ia ion in he Displacing Solu ion . . . . . . . . . . . . . . 127
8.3 Quan i a i eAnalysis............................... 128
8.3.1 In e ace Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.3.2 Ci cula i y Va ia ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.4 Ins abili y Mechanism - Schlie en Expe imen s . . . . . . . . . . . . . . . . . 131
8.5 Expe imen al Damkh¨
ole Numbe s........................ 133
8.6 Chap e Discussion ................................ 134
9 Nume ical Resul s 135
9.1 Ba ch Sys ems Simula ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.1.1 FS-PAASys em ............................. 135
9.1.2 FSG-PAASys em ............................ 137
9.2 Non-Linea DC Simula ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.3 In e aceThickness................................ 138
9.4 Ci cula i yVa ia ion ............................... 139
9.4.1 Dependence on [Fo maldehyde]0.................... 139
9.4.2 Dependence on [SO32 – ]0......................... 140
9.5 E ec s o he Reagen Concen a ion on he Spa ial Viscosi y P o iles . . . . . 141
9.6 E ec o Di usion on he Sys em Beha io . . . . . . . . . . . . . . . . . . . 142
9.7 Chap e Discussion ................................ 143
Pa II Conclusions 145
Pa III: Complex Pa e n Fo ma ion and Viscous Finge ing
S abiliza ion 149
Pa III Mo i a ion 149
10 Expe imen al and Nume ical Me hods 151
10.1Expe imen alMe hods .............................. 151
10.1.1 ChemicalRecipes............................. 151
10.1.2 Expe imen al Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.1.3 Radial Hele-Shaw Cell: Expe imen s and P o ocols . . . . . . . . . . . 152
10.1.4 Schlie enImaging ............................ 153
10.1.5 Con ol Expe imen s . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
10.1.6 P ecipi a ion ............................... 154
10.1.7 Scanning Elec on Mic oscopy (SEM) . . . . . . . . . . . . . . . . . . 155
10.2Nume icalMe hods................................ 155
10.2.1 Non-Linea Reac ion-Di usion-Con ec ion (RDC) Simula ions . . . . 155
10.2.2 Mesh Independence S udy Fo The RDC Model . . . . . . . . . . . . . 157
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11 Expe imen al Resul s 159
11.1CaseI ....................................... 159
11.1.1 Gene al Sys em O e iew . . . . . . . . . . . . . . . . . . . . . . . . 159
11.1.2 Desc ip i e Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
11.1.3 Quan i a i e Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
11.2CaseII....................................... 164
11.2.1 Gene al Sys em O e iew . . . . . . . . . . . . . . . . . . . . . . . . 164
11.2.2 Desc ip i e Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11.2.3 Quan i a i e Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
11.3Chap e Discussion ................................ 168
12 Chemical Analysis and Ins abili y Mechanism 169
12.1 Expe imen al Analysis - Con ol Expe imen s . . . . . . . . . . . . . . . . . . 169
12.1.1 C1: In luence o he Colo Indica o . . . . . . . . . . . . . . . . . . . 169
12.1.2 C2: In luence o he Fo maldehyde . . . . . . . . . . . . . . . . . . . 169
12.1.3 C3: In luence o he SO32 – ........................ 170
12.1.4 C4: In luence o he PAA . . . . . . . . . . . . . . . . . . . . . . . . . 170
12.1.5 C5: Chemical In e ac ion a he In e ace . . . . . . . . . . . . . . . . 171
12.2 Physical and Chemical Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 174
12.2.1 C us Fo ma ion ............................. 174
12.2.2 Reac i eF on .............................. 177
12.2.3 Reac ion F on Veloci y and Damhk¨
ole Numbe Calcula ion . . . . . . 178
12.2.4 P´
ecle -Damhk¨
ole Numbe (PeDa).................... 179
12.3Chap e Discussion ................................ 181
13 Nume ical Model and Resul s 183
13.1 Reac ion-Di usion-Con ec ion (RDC) Nume ical Model . . . . . . . . . . . . 183
13.2 Nume ical Resul s o Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.2.1 Sys emDynamics ............................ 185
13.2.2 E ec o he Flow Ra e . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.2.3 Ci cula i y Calcula ion . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.3 Nume ical Resul s o Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
13.3.1 Sys emDynamics ............................ 187
13.3.2 E ec o he Flow Ra e . . . . . . . . . . . . . . . . . . . . . . . . . . 188
13.3.3 Ci cula i y Calcula ion . . . . . . . . . . . . . . . . . . . . . . . . . . 188
13.3.4 DisplacedVolume ............................ 189
13.4Supplemen a yResul s .............................. 190
13.4.1 E ec o he Di usion . . . . . . . . . . . . . . . . . . . . . . . . . . 190
13.4.2 E ec o Va ying K ........................... 190
13.5Chap e Discussion ................................ 192
Pa III Conclusions 195
Gene al Conclusions 197
Appendices 201

Con en s
Appendix A: P epa a ion o S ock Solu ions 201
A.1 S ock Solu ions Used Fo Pa I . . . . . . . . . . . . . . . . . . . . . . . . . . 201
A.1.1 CHDSolu ion .............................. 201
A.1.2 Sodium B oma e Solu ion . . . . . . . . . . . . . . . . . . . . . . . . 201
A.1.3 Fe oinSolu ion ............................. 201
A.1.4 Sodium Sul a e Solu ion . . . . . . . . . . . . . . . . . . . . . . . . . 201
A.1.5 Sul u ic Acid Solu ion . . . . . . . . . . . . . . . . . . . . . . . . . . 202
A.1.6 Sodium Chlo ide Solu ion . . . . . . . . . . . . . . . . . . . . . . . . 202
A.1.7 Reac i e Mix u e: Solu ions 1 and 2 . . . . . . . . . . . . . . . . . . . 202
A.1.8 P o ocol o p epa e he BZ-Aga ose Gels . . . . . . . . . . . . . . . . 202
A.2 S ock Solu ions Used o Pa II . . . . . . . . . . . . . . . . . . . . . . . . . 203
A.2.1 Fo maldehyde Solu ion . . . . . . . . . . . . . . . . . . . . . . . . . . 203
A.2.2 Sodium Sul i e Solu ion . . . . . . . . . . . . . . . . . . . . . . . . . 203
A.2.3 Poly(Ac ylic Acid) [PAA] solu ions . . . . . . . . . . . . . . . . . . . 203
A.2.4 Sodium Bisul i e Solu ion . . . . . . . . . . . . . . . . . . . . . . . . 203
A.2.5 Sodium Hyd oxide Solu ion . . . . . . . . . . . . . . . . . . . . . . . 203
A.2.6 Gluconolac one Solu ion . . . . . . . . . . . . . . . . . . . . . . . . . 203
A.2.7 B omo hymol Blue Indica o . . . . . . . . . . . . . . . . . . . . . . . 203
A.2.8 Displacing Solu ion Mix u e P epa a ion . . . . . . . . . . . . . . . . 204
A.3 S ock Solu ions Used o Pa III . . . . . . . . . . . . . . . . . . . . . . . . . 204
A.3.1 Fo maldehyde Solu ion . . . . . . . . . . . . . . . . . . . . . . . . . . 204
A.3.2 Sodium Sul i e Solu ion . . . . . . . . . . . . . . . . . . . . . . . . . 204
A.3.3 Poly(Ac ylic Acid) [PAA] solu ion . . . . . . . . . . . . . . . . . . . . 204
A.3.4 Sodium Ca bona e Solu ion . . . . . . . . . . . . . . . . . . . . . . . 204
A.3.5 B omo hymol Blue Indica o . . . . . . . . . . . . . . . . . . . . . . . 204
A.3.6 Gluconic Acid Solu ion (Solu ion B) . . . . . . . . . . . . . . . . . . . 204
A.3.7 Solu ion A Mix u e P epa a ion . . . . . . . . . . . . . . . . . . . . . 204
Appendix B: Reac ion-Di usion Sys ems 205
B.1 Ma e ialsandMe hods .............................. 205
B.1.1 Capilla ySys em............................. 205
B.1.2 Non-Con ec i e Aga ose-Based Sys em . . . . . . . . . . . . . . . . . 206
B.2 Nume icalModels ................................ 207
B.2.1 Go e ning equa ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
B.2.2 1D-RD Sys em Model Se up . . . . . . . . . . . . . . . . . . . . . . . 208
B.2.3 2D-RD Sys em Model Se up . . . . . . . . . . . . . . . . . . . . . . . 208
B.3 Expe imen alResul s............................... 209
B.3.1 1D Capilla y Sys em . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.3.2 Non-Con ec i e Aga ose-Based Sys em . . . . . . . . . . . . . . . . . 209
B.4 Nume icalResul s................................. 212
B.4.1 1.5D-RD Simula ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
B.4.2 2D-RDSimula ions............................ 212
B.5 Discussion..................................... 216
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Appendix C: Image Analysis Techniques 217
C.1 Analysis Me hods Used o Pa I . . . . . . . . . . . . . . . . . . . . . . . . . 217
C.1.1 Calcula ion o Measu ing Obse ables . . . . . . . . . . . . . . . . . . 217
C.2 Analysis Me hods Used o Pa II . . . . . . . . . . . . . . . . . . . . . . . . 218
C.2.1 Ci cula i y Calcula ion . . . . . . . . . . . . . . . . . . . . . . . . . . 218
C.2.2 Densi y A ea Calcula ion . . . . . . . . . . . . . . . . . . . . . . . . . 219
C.3 Analysis Me hods Used o Pa III . . . . . . . . . . . . . . . . . . . . . . . . 220
C.3.1 Time-Dependen Ci cula i y Calcula ion . . . . . . . . . . . . . . . . . 220
C.3.2 A e age Displacing P o ile . . . . . . . . . . . . . . . . . . . . . . . . 220
Appendix D: Supplemen a y Resul s 223
D.1 Resul s o Pa I ................................. 223
D.1.1 Solu al Expansion Coe icien Calcula ion . . . . . . . . . . . . . . . . 223
D.1.2 RDC Model Pe meabili y Va ia ion . . . . . . . . . . . . . . . . . . . 223
D.1.3 RDC Con ol Simula ion . . . . . . . . . . . . . . . . . . . . . . . . . 223
D.2 Resul s o Pa II................................. 226
D.2.1 E ec o he Colo Indica o . . . . . . . . . . . . . . . . . . . . . . . 226
D.2.2 Elas ici y E ec s and Shea Ra e Es ima ion . . . . . . . . . . . . . . . 226
D.3 Resul s o Pa III ................................ 228
D.3.1 Shea Ra e Es ima ion . . . . . . . . . . . . . . . . . . . . . . . . . . 228
D.3.2 P´
ecle Numbe (Pe)............................ 228
Appendix E: Copy igh Pe missions 231
Bibliog aphy 235
ii
Summa y
This hesis p esen s he wo k ealized in he Non-Linea Physics G oup o he Uni e si y
o San iago de Compos ela. This wo k in oduces an in e disciplina y s udy o sys ems c ea ed
om coupling hyd odynamic ins abili ies in Hele-Shaw cells and complex chemical eac ions.
Objec i e and S a e o he A
The main objec i e o his hesis is o ind a nexus be ween wo di e en ields: On he one
hand, he hyd odynamic ins abili ies, and on he o he hand, he ex ensi e wo ld o complex
chemical eac ions. Inside he ield o hyd odynamics, his wo k will s udy a mo e speci ic
ype o ins abili ies known as inge ing ins abili ies. These ins abili ies occu when a luid wi h
low mobili y is displaced by a luid wi h high mobili y [45, 44, 93, 168]. The mobili y can
be a ec ed by he iscosi y, densi y, su ace ension, o empe a u e o he luid. The main
cha ac e is ic o hese ins abili ies is ha hey exhibi well-de ined pa e ns simila o inge s,
ha occu du ing he luid displacemen [45].
The inge ing phenomenon was obse ed in many ields o science and indus y. Some
examples a e:
• Enhanced Oil Reco e y (EOR). In his case, inge ing occu s du ing he oil ex ac ion
p ocess, when wa e is used o displace he mo e iscous oil. This phenomenon di ec ly
a ec s he e iciency o he displacemen and i is de imen al o he eco e y p ocess
[132, 206, 41, 21, 207, 210].
• Hyd ogeological phenomena. Many s udies p edic he occu ence o saline inge s
wi hin he anspo o subsu ace aqui e s. These ypes o inge s a e obse ed when
he colde and mo e saline seawa e makes con ac wi h esh wa e . The di e ences in
salini y and empe a u e p oduce a cha ac e is ic inge ing known as sal inge s. This
p ocess a o s nu ien oxida ion in eshwa e anspo [111, 200, 171, 76].
• In ch oma og aphic p ocesses. In his sepa a ion echnique, he componen s o a mix u e
a e sepa a ed when he solu ion passes h ough a po ous ma ix. I has been demons a ed
he occu ence o iscous inge ing when a iscous sample is elu ed using a less iscous
sol en . Simila o he EOR case, his p ocess is de imen al o he ex ac ion [173, 32,
42, 198, 166].
Howe e , his wo k will be mo e ocused on hose ins abili ies p oduced in expe imen al
de ices known as Hele-Shaw cells [89]. These cells a e made o wo pa allel pla es sepa a ed
by a e y small gap. This de ice allows s udying he inge ing phenomenon in a con olled and
Resumen
• En los p ocesos c oma og ´
a icos. La c oma og a ´
ıa de exclusi´
on molecula es una ´
ecnica
de sepa aci´
on ampliamen e u ilizada en la qu´
ımica, biolog´
ıa molecula , bio ecnolog´
ıa,
en e o os. Es un p oceso median e el cual los solu os de una mezcla son sepa ados po
di e encias de ama˜
no, al pasa la misma po un medio po oso. Se ha demos ado, que
al que e elui una mues a iscosa con un sol en e menos iscoso se p oduce digi aci´
on
iscosa que a ec a nega i amen e an o a la ex acci´
on como a los endimien os [173, 32,
42, 198, 166].
Sin emba go, es e abajo se en oca en el es udio de las ines abilidades p oducidas en
disposi i os expe imen ales conocidos como celdas de Hele-Shaw [89]. Es os disposi i os es ´
an
con o madas po dos placas sepa adas es echamen e una de la o a, pe mi iendo el es udio del
p oceso de digi aci´
on de una mane a ela i amen e sencilla y con olada. Poseen adem´
as la
pa icula idad de que el lujo den o de ellas es id´
en ico al de un luido a a ´
es de un medio
po oso, po lo que los es udios lle ados a cabo en ellas pueden se ep esen a i os de sis emas
muchos m´
as complejos y de pa icula in e ´
es en la indus ia.
El en´
omeno de digi aci´
on en celdas de Hele-Shaw ue y sigue siendo ex ensamen e
es udiado y ca ac e izado desde hace d´
ecadas . Los p ime os es udios abo da on la p oblem´
a ica
de desplazamien os no eac i os an o miscibles como inmiscibles, demos ando la impo ancia
de ac o es como la ensi´
on supe icial o la di usi idad en la deses abilizaci´
on de sis emas
[93, 48, 127, 197, 8, 45].
M´
as ecien emen e, se es udia on los p ocesos mediados po eacciones qu´
ımicas, en donde
sis emas con mo ilidades iniciales a o ables (y po lo an o, es ables), son deses abilizados po
p ocesos isicoqu´
ımicos acoplados al desplazamien o [45, 25, 44, 9].
El es udio de eacciones qu´
ımicas acopladas al en´
omeno de digi aci´
on ue inc emen ando
su complejidad con el iempo. As´
ı, eacciones elemen ales como p ocesos de neu alizaci´
on
o de p ecipi aci´
on qu´
ımica, ue on y son hoy en d´
ıa ex ensamen e ca ac e izados mos ando
un ico abanico de compo amien os. El e ec o de es as eacciones en las ines abilidades es ´
a
mediado p incipalmen e po el cambio en las azones de mo ilidad debido al inc emen o de
ac o es como la densidad o la iscosidad de los luidos in oluc ados o la pe meabilidad del
medio [141, 138, 137, 139, 73, 65, 9, 176].
La complejidad de los p ocesos qu´
ımicos acoplados al desa ollo de ines abilidades puede
inc emen a se conside ando lo que denominamos eacciones qu´
ımicas complejas. En un
con ex o gene al, es as eacciones son de inidas como p ocesos qu´
ımicos que se p oducen en
a ias e apas (al menos dos) y cuen an con al menos un in e media io de eacci´
on [116]. Den o
de es a de inici´
on, se engloba un campo muy ex enso de eacciones. Algunas de es as, ya han
sido acopladas a ines abilidades con an e io idad, como po ejemplo los p ocesos au oca al´
ı icos
[101, 49, 96].
En el con ex o de es e abajo, u iliza emos eacciones qu´
ımicas cuyos mecanismos apo an
un paso m´
as de complejidad espec o a las eacciones au oca al´
ı icas y donde los in e media ios
de eacci´
on desempe˜
nan un papel undamen al pa a el desa ollo de la din´
amica de las mismas.
As´
ı pues, en es e abajo, se analiza ´
a el acople de las ines abilidades hid odin´
amicas con dos
ipos de eacciones complejas: La eacci´
on oscilan e de Belouso -Zhabo insky (BZ) [18] y las
eacciones de cambio de pH [106, 108, 107, 159, 160, 145].
La eacci´
on BZ, es uno de los oscilado es qu´
ımicos m´
as ex ensamen e conocidos. Es
un p oceso edox en donde los in e media ios de eacci´
on oscilan en el iempo median e la
din´
amica de ac i ado /inhibido [82]. Es as oscilaciones se ap ecian con cambios de colo
pe i´
odicos en el caso de un sis ema pe ec amen e agi ado o con la o maci´
on de pa ones
x i

DAR´
IO MART´
IN ESCALA VODOPIVEC
espacio empo ales en sis emas espacialmen e ex endidos. Exis en a iaciones de es a eacci´
on.
En es e abajo en pa icula se op ´
o po usa una eacci´
on conocida como eacci´
on BZ-CHD,
en donde el sus a o o g´
anico de la ece a o iginal se eemplaza po el componen e CHD
(1,4-Ciclohexanodiona) [113, 112, 187, 188, 190, 114]. Es a o mulaci´
on p esen a a ias
en ajas espec o a la ece a o iginal, pe o la m´
as signi ica i a es que no p oduce di´
oxido de
ca bono como p oduc o secunda io, lo cual es undamen al si se p e ende acopla es a eacci´
on
a en´
omenos hid odin´
amicos en medios con inados.
Po o a pa e, las eacciones de cambio de pH son eacciones en donde el pH del sis ema
cambia de un es ado b´
asico (o ´
acido) a un es ado ´
acido (o b´
asico) debido a la in e acci´
on de
los in e media ios de eacci´
on [107, 108, 159, 153, 145]. Exis en eacciones de cambio de
pH ino g´
anicas, las cuales son ambi´
en p ocesos edox y pueden mos a oscilaciones an o
en sis emas abie os como en sis emas semi abie os. Es os sis emas han sido ex ensamen e
ca ac e izados habiendo una g an lis a de o mulaciones disponibles [145]. Sin emba go, debido
a su na u aleza edox, es os sis emas no pueden acopla se a elemen os o g´
anicos, como son los
pol´
ıme os o sis emas de libe aci´
on de ´
a macos (o d ug deli e y po su nomb e de ingl´
es).
Recien emen e, se desa olla on eacciones de cambio de pH o g´
anicas que a di e encia
de las an e io es no son p ocesos edox, y po lo an o son menos ag esi as con elemen os
o g´
anicos como los an es indicados [106, 108, 107]. Es as eacciones ambi´
en mues an
compo amien os oscila o ios en sis emas abie os y han sido ela i amen e bien ca ac e izadas.
Sin emba go, debido a no se eacciones del ipo edox, hay pocas o mulaciones disponibles.
En es e abajo se u iliza ´
an dos eacciones de cambio de pH o g´
anicas,
la eacci´
on de Fo maldehido-Sul i o (FS) [212, 106, 213] y la eacci´
on de
Fo maldeido-Sul i o-Gluconolac ona (FSG) [108, 107]. La p ime a es ambi´
en conocida
como eacci´
on Clock, po que el pH de la soluci´
on cambia en un momen o de e minado
de ´
acido a b´
asico. El iempo de cambio depende de las concen aciones de las especies
in oluc adas. La segunda eacci´
on, es una de i aci´
on de la p ime a en donde se acopla la
hid ´
olisis de la gluconolac ona al cambio de pH de la eacci´
on de Sul i o-Fo maldehido.
As´
ı pues, el obje i o de es udio de es a esis es acopla en e si odos los en´
omenos
desc i os an e io men e. De mane a espec´
ı ica se es udia ´
an los siguien es sis emas:
•Sis ema 1: Acople en e la eacci´
on BZ-CHD y una ines abilidad de digi aci´
on inducida
po e ec o de la lo abilidad en una celda o ien ada e icalmen e.
•Sis ema 2: Acople en e la eacci´
on FS y una ines abilidad de digi aci´
on iscosa en una
celda o ien ada ho izon almen e.
•Sis ema 3: Acople en e la eacci´
on FSG y una ines abilidad de digi aci´
on iscosa en una
celda o ien ada ho izon almen e.
De es a mane a se p e ende cons ui es os sis emas expe imen almen e y abaja con
ellos en un ango de pa ´
ame os en donde los iempos ca ac e ´
ıs icos sean compa ables y
ocu an sine gia en e ambos en´
omenos. Se ealiza ´
a la comple a ca ac e izaci´
on, se plan ea ´
an
hip´
o esis espec o a los mecanismos de in e acci´
on in oluc ados y se es ablece ´
an modelos
ma em´
a icos que pe mi an simula la din´
amica de los mismos.
x ii
Resumen
Es uc u a de la esis
Es a abajo es ´
a di idido en es pa es. Cada pa e cuen a con una in oducci´
on, un
cap´
ı ulo de m´
e odos expe imen ales y num´
e icos, un cap´
ı ulo de esul ados expe imen ales,
un cap´
ı ulo de esul ados num´
e icos y una discusi´
on gene al. Independien emen e del hilo
conduc o com´
un que poseen las pa es, se eligi´
o o ganiza el abajo de es a o ma no solo
pa a acili a su lec u a, sino adem´
as po que cada pa e en s´
ı posee su icien e con enido
independien e que jus i ica el uso del o ma o.
De es a mane a, los Cap´
ı ulos 2,6y10, son cap´
ı ulos que desa ollan los m´
e odos
an o expe imen ales como num´
e icos de las Pa es I,II yIII espec i amen e. En
la secci´
on expe imen al de es os cap´
ı ulos se encuen an odas las ece as, o mulaciones
qu´
ımicas y p o ocolos u ilizados en los expe imen os, as´
ı como la desc ipci´
on de los a eglos
expe imen ales y es udios p e ios necesa ios pa a el desa ollo del abajo. En la secci´
on
num´
e ica, se incluyen conside aciones gene ales, como la desc ipci´
on de los dominios
compu acionales, con igu aci´
on de los p og amas empleados en las simulaciones, pa ´
ame os
u ilizados, en e o os aspec os. Sin emba go y pa a acili a la lec u a de la esis, el desa ollo
de las ecuaciones undamen ales de los modelos se ese a pa a los co espondien es cap´
ı ulos
de esul ados num´
e icos, ya que en la mayo ´
ıa de los casos, ´
es os dependen de los esul ados
expe imen ales.
Los concep os b´
asicos son in oducidos en el Cap´
ı ulo 1. En es e cap´
ı ulo se explican
una se ie de cues iones undamen ales pa a el en endimien o del abajo. As´
ı, se desc iben las
eacciones oscilan es cl´
asicas, como la eacci´
on de Belouso -Zhabo insky y los oscilado es
de pH an o ino g´
anicos como los o g´
anicos. Tambi´
en se in oducen las ines abilidades
hid odin´
amicas en celda Hele-Shaw y los en´
omenos de digi aci´
on po di e encia de iscosidad
y de densidad. Complemen a iamen e, se incluye una secci´
on pa a desc ibi las p opiedades
m´
as impo an es del ´
acido poliac ´
ılico, que es un componen e ampliamen e u ilizado en el
desa ollo de es e abajo, m´
as p ecisamen e en las Pa es II yIII. Finalmen e, se desc ibe
la ´
ecnica ´
op ica de Schlie en. Es a es una ´
ecnica de isualizaci´
on de luidos ampliamen e
u ilizada du an e odo el desa ollo de la esis y que ue undamen al pa a el en endimien o de
los mecanismos ´
ısicos y qu´
ımicos de los en´
omenos obse ados expe imen almen e.
La Pa e I comienza inmedia amen e despu´
es del cap´
ı ulo in oduc o io. En es a
pa e se es udia el acople en e la eacci´
on BZ-CHD (una a iaci´
on de la eacci´
on
Belouso -Zhabo insky), y la ines abilidad hid odin´
amica de digi aci´
on inducida po lo abilidad
en una celda Hele-Shaw o ien ada e icalmen e.
En el Cap´
ı ulo 3 se p esen an los esul ados expe imen ales ob enidos pa a es e sis ema.
Los mismos son analizados de o ma an o desc ip i a como cuan i a i a, es udiando los e ec os
p oducidos po los cambios en la densidad (∆ρ) y la exci abilidad (ε). Es os pa ´
ame os
ue on modi icados an o de mane a desacoplada como acoplada, y su e ec o ue ca ac e izado
median e el es udio de obse ables espec´
ı icos como los iempos de inducci´
on qu´
ımicos ( ind−C)
e hid odin´
amicos ( ind−H), la longi ud de onda (λH) y el pe ´
ıodo (TC). De es a o ma, se p esen a
un es udio de ca ac e izaci´
on comple o de los en´
omenos obse ados, pudi´
endose es ablece las
p incipales in luencias de la hid odin´
amica y la qu´
ımica en la din´
amica del sis ema.
En el Cap´
ı ulo 4 se p esen a un es udio qu´
ımico de allado ealizado con la inalidad de
descub i el mecanismo de ´
as de los esul ados expe imen ales en el Cap´
ı ulo 3. Dicho es udio
se lle ´
o a cabo median e una ex ensi a se ie de expe imen os de con ol y ca ac e izaci´
on
qu´
ımica, en donde se u iliza on di e sas he amien as de an´
alisis ales como espec o o ome ´
ıa
x iii
DAR´
IO MART´
IN ESCALA VODOPIVEC
y esonancia magn´
e ica nuclea . Median e es os es udios, es iden i icado el ac o p incipal en
el acople en e la eacci´
on BZ-CHD y la ines abilidad hid odin´
amica.
En el Cap´
ı ulo 5, se aplican los esul ados ob enidos en el Cap´
ı ulo 4 en combinaci´
on
con los modelos cin´
e icos ya exis en es, pa a ep oduci de o ma na u al los esul ados
expe imen ales median e simulaciones. As´
ı, en es e cap´
ı ulo se compa an p ime amen e los
esul ados expe imen ales y los modelos cin´
e icos con la inalidad de ob ene un ango de
pa ´
ame os de abajo equi alen e. Luego, se p opone un modelo num´
e ico ex ensi o de
eacci´
on-di usi´
on-con ecci´
on incluyendo los mecanismos descubie os y desa ollando paso a
paso odas las ecuaciones undamen ales. Finalmen e, los esul ados num´
e icos son analizados
an´
alogamen e a los expe imen ales demos ando las equi alencias en e ambos sis emas.
La Pa e II comienza inmedia amen e luego del Cap´
ı ulo 5. En es e caso se es udia el
acople en e una ines abilidad hid odin´
amica de digi aci´
on iscosa y una eacci´
on de cambio de
pH en una celda de Hele-Shaw o ien ada ho izon almen e.
P e iamen e al es udio del sis ema hid odin´
amico, es necesa io desa olla un sis ema
qu´
ımico en donde la iscosidad sea a ec ada po cambios en el pH. De es a mane a, en el
Cap´
ı ulo 7 se desa olla un sis ema en donde se obse an expe imen almen e a iaciones
de empo ales sin´
e gicas en e el pH y la iscosidad. Es o se log a acoplando un pol´
ıme o
sensible al pH, (el ´
acido poliac ´
ılico), con dos eacciones de cambio de pH: la eacci´
on
de Fo maldehido-Sul i o y la eacci´
on de Fo maldehido-Sul i o-Gluconolac ona. Ambos
sis emas son ex ensamen e ca ac e izados median e ´
ecnicas eol´
ogicas y anal´
ı icas, midiendo
obse ables como los sal os de iscosidad y pH (∆µy∆pH espec i amen e), los cambios en
la din´
amica empo al espec o a la o mulaci´
on o iginal y los iempos de inducci´
on del sis ema.
Los obse ables ue on es udiados pa a cada especie qu´
ımica in oluc ada en las o mulaciones
u ilizadas. Finalmen e, la ´
ul ima pa e de es e cap´
ı ulo demues a el mecanismo de acople
median e expe imen os de con ol y u as al e na i as conside ando o mulaciones di e sas.
En el Cap´
ı ulo 8, los desa ollos ob enidos en el cap´
ı ulo an e io son adap ados a una
celda de Hele-Shaw pa a gene a , con ´
exi o, una ines abilidad de digi aci´
on iscosa. Es e
sis ema se es udia de mane a desc ip i a analizando los e ec os p oducidos po los pa ´
ame os
m´
as impo an es como son el caudal olum´
e ico (Q) y la composici´
on de las o mulaciones
u ilizadas. Seguidamen e el sis ema se ca ac e iza cuan i a i amen e, es udiando la mo olog´
ıa
de los pa ones obse ados median e el c´
alculo de la ci cula idad (C). Tambi´
en se u iliza la
´
ecnica de Schlie en como complemen o pa a dilucida el mecanismo de la ines abilidad.
El Cap´
ı ulo 9 p esen a los esul ados num´
e icos de los sis emas p esen ados en los
Cap´
ı ulos 7 y8. En la p ime a pa e, se adap an los modelos cin´
e icos exis en es de las
eacciones de pH a los expe imen os. Es o se ealiza encon ando el conjun o de pa ´
ame os
que mejo ajus a los modelos a los esul ados expe imen ales. Los modelos ajus ados son
analizados de mane a an´
aloga a como se hace en el Cap´
ı ulo 7. Seguidamen e, se p esen an
las ecuaciones undamen ales de un modelo num´
e ico de con ecci´
on-di usi´
on pa a simula el
sis ema in oducido en el Cap´
ı ulo 8. Los esul ados son analizados midiendo la ci cula idad
num´
e ica de la misma o ma que se hizo en los expe imen os. Se demues a que an o los
esul ados expe imen ales como los num´
e icos concue dan m´
as que acep ablemen e.
Seguidamen e comienza la Pa e III en donde se es udia el acople en e la eacci´
on
de Sul i o-Fo maldehido-Gluconolac ona y el en´
omeno de digi aci´
on iscosa en celda de
Hele-Shaw o ien ada ho izon almen e. En es a pa e se es udian dos si uaciones p o enien es de
un mismo sis ema expe imen al. Es os casos se ob ienen in e cambiando solamen e la soluci´
on
desplazan e po la desplazada. En el llamado Caso I, se analiza la gene aci´
on de pa ones
xix
Resumen
complejos en condiciones de apa en e es abilidad hid odin´
amica. En el Caso II se analiza
la es abilizaci´
on de digi aci´
on iscosa median e p ocesos isicoqu´
ımicos. Si bien el sis ema
expe imen al es udiado en es a pa e es de i ado de la Pa e II, la iqueza, e sa ilidad y
complejidad de los esul ados ob enidos han hecho necesa io dedica un espacio apa e pa a
ellos.
As´
ı, el Cap´
ı ulo 11 es udia la din´
amica de los dos casos conside ados. En ambas
si uaciones, se ealiza on es udios desc ip i os seguidos de es udios cuan i a i os pa a casos
eac i os y no eac i os. Los es udios cuan i a i os se ealiza on mayo men e midiendo la
ci cula idad, siendo posible demos a una ue e dependencia de los esul ados con la elocidad
de desplazamien o y la composici´
on qu´
ımica de los sis emas. En ambos casos, el caudal
olum´
e ico ue elegido como el pa ´
ame o de an´
alisis undamen al, siendo es e a iado en
un amplio ango de alo es y pe mi iendo la ca ac e izaci´
on comple a del sis ema.
El Cap´
ı ulo 12 se cen a en encon a un mecanismo que d´
e explicaci´
on a los en´
omenos
obse ados. Es o se lle a a cabo median e un ex ensi o an´
alisis con expe imen os de con ol,
en donde las o mulaciones qu´
ımicas ue on modi icadas minuciosamen e pa a pode en ende
el e ec o p oducido po cada uno de las especies en el sis ema. Bas´
andose en es os esul ados,
se p opone un modelo qu´
ımico y se ealizan c´
alculos complemen a ios pa a undamen a lo.
Finalmen e, en el Cap´
ı ulo 13 se p esen a un modelo num´
e ico de
eacci´
on-di usi´
on-con ecci´
on que simula los en´
omenos desc i os en el Cap´
ı ulo 11, en
donde se de allan paso po paso el desa ollo de las ecuaciones undamen ales del mismo. Los
esul ados num´
e icos se analizan una ez m´
as imi ando el an´
alisis ealizado a los esul ados
expe imen ales, demos ando una g an conco dancia en e los mismos. El modelo ambi´
en se
u iliza como he amien a pa a explica el e ec o de la di usi´
on en la din´
amica de los casos de
es udio.
Una ez inalizada la e ce a pa e, se p esen a una se ie de ap´
endices con in o maci´
on
complemen a ia.
En el Ap´
endice A se mues an odas las ece as de las soluciones mad es u ilizadas pa a
lle a a cabo es e abajo. En las mismas, se de allan e e encias, p o ocolos y can idades de los
qu´
ımicos u ilizados. Tambi´
en se de allan las ece as de las soluciones eac i as usadas en cada
pa e.
En el Ap´
endice B, se p esen an los esul ados, an o num´
e icos como expe imen ales, de los
sis emas de eacci´
on-di usi´
on de i ados de la Pa e I. Es os esul ados ue on impo an es pa a
el desa ollo y el en endimien o p e io del mecanismo p opues o en dicha pa e. Sin emba go,
se ha decidido inclui lo en un ap´
endice como esul ados complemen a ios. En es e ap´
endice se
es udian sis emas capila es en 1D y con ma iz de aga osa en 2D.
En el Ap´
endice C se p esen an los m´
e odos y p o ocolos u ilizados pa a el a amien o de
im´
agenes expe imen ales y num´
e icas. Es os m´
e odos son undamen ales pa a la ob enci´
on de
los alo es cuan i a i os p esen ados a lo la go del abajo. El ap´
endice es ´
a di idido en los
m´
e odos u ilizados en cada pa e.
En el Ap´
endice D se incluye in o maci´
on y c´
alculos complemen a ios que ue on u ilizados
pa a el desa ollo de los cap´
ı ulos p incipales, pe o que no son incluidos di ec amen e en el
cue po de la esis debido a que su apo e no es especialmen e signi ica i o.
Finalmen e, el Ap´
endice E mues a los pe misos pa a el uso, ep oducci´
on, modi icaci´
on y
publicaci´
on del ma e ial pe enecien e a aquellas publicaciones en donde los de echos de au o
ue on cedidos. Es os pe misos se incluyen pa a e i a cualquie p oblema legal e e en e al
plagio o au oplagio de in o maci´
on.
xx
Resumo
A p esen e memo ia esume o aballo ealizado no G upo de F´
ısica Non Lineal da USC.
Na mesma exponse o es udo an o expe imen al como num´
e ico de sis emas xe ados median e
a combinaci´
on en e as ines abilidades hid odin´
amicas en c´
elulas de Hele-Shaw e eacci´
ons
qu´
ımicas complexas
Obxec i o e Es ado do A e
O obxec i o des a ese ´
e o de a opa un nexo com´
un en e dous mundos apa en emen e moi
di e en es. Po unha banda, o campo das ines abilidades hid odin´
amicas e po ou a, o campo
do que nes e con ex o denominamos eacci´
ons qu´
ımicas complexas. Como odo aballo de
especializaci´
on, ´
e undamen al de ini conc e amen e os l´
ımi es do es udo. Is o ´
e a´
ında m´
ais
necesa io, endo en con a que ´
ambolos dous mundos son ex emadamen e ex ensos
Den o do campo das ines abilidades hid odin´
amicas, nes e aballo es ud´
asense un ipo
pa icula de ines abilidades co˜
necidas como dixi aci´
ons (ou inge ing en ingl´
es). Es as
ines abilidades obs´
e anse cando un lu´
ıdo con ce as p opiedades e modin´
amicas (como a
empe a u a, iscosidade, densidade ou a ensi´
on supe icial), en a en con ac o con ou o, e
desp ´
azao [45, 44, 93, 168]. Se as condici´
ons de desp azamen o son des a o ables, o a as e
p od´
ucese de manei a non homox´
enea, xe ando pa ´
ons simila es a dedos (de a´
ı o nome
de dixi aci´
on) [45]. Es es sis emas son conside ados ines ables desde o pun o de is a da
luidodin´
amica.
O en´
omeno de dixi aci´
on obse ouse amplamen e en moi os campos da ciencia e a
indus ia, as´
ı po nomea alg´
uns poden ci a se:
• A ex acci´
on mello ada de pe ´
oleo ou Enhanced Oil Reco e y-EOR polas s´
uas siglas
en ingl´
es. Nes e caso, o en´
omeno de dixi aci´
on iscosa oco e ao que e desp aza o
pe ´
oleo (un lu´
ıdo cunha maio iscosidade) median e a inxecci´
on de auga a p esi´
on
(un lu´
ıdo cunha iscosidade meno ). O desp azamen o ines able a ec a di ec amen e
o p oceso de ex acci´
on, acendo que es e sexa ine icien e, e po an o p oducindo uns
endemen os educidos [132, 206, 41, 21, 207, 210].
• En en´
omenos hid oxeol´
oxicos: Va ios es udos p ed´
ın a apa ici´
on de dixi aci´
ons salinas
no anspo e de acu´
ı e os sub e ´
aneos. Es e ipo de dixi aci´
ons sucede cando a auga do
ma , que pos´
ue unha concen aci´
on salina e empe a u a ca ac e ´
ıs ica, en a en con ac o
con auga doce. As di e enzas de salinidade e empe a u a p oducen dixi aci´
ons co˜
necidas
como dedos de sal (ou sal inge s en ingl´
es) que in e i en nega i amen e nas desca gas
de auga doce, a o ecendo a oxidaci´
on de nu ien es [111, 200, 171, 76].

Resumo
• Nos p ocesos c oma og ´
a icos. A c oma og a ´
ıa de exclusi´
on molecula ´
e unha ´
ecnica
de sepa aci´
on amplamen e u ilizada na qu´
ımica, biolox´
ıa molecula , bio ecnolox´
ıa, en e
ou os. ´
E un p oceso median e o cal una os solu os dunha mes u a son sepa ados po
di e enzas de ama˜
no, ao pasa a mesma po un medio po oso. Demos ouse, que ao
que e elui unha mos a iscosa cun sol en e menos iscoso p od´
ucese un en´
omeno
de dixi aci´
on iscosa que a ec a nega i amen e an o ´
a ex acci´
on como aos endemen os
[173, 32, 42, 198, 166].
Con odo, es e aballo en ´
ocase no es udo das ines abilidades p oducidas en disposi i os
expe imen ais co˜
necidos como c´
elulas de Hele- Shaw [89]. Es es disposi i os es ´
an
con o madas po d´
uas placas sepa adas es ei amen e una da ou a, pe mi indo o es udo do
p oceso de dixi aci´
on dunha manei a ela i amen e sinxela e con olada. Pos´
uen ademais a
pa icula idade de que o luxo den o delas ´
e id´
en ico ao dun lu´
ıdo a a ´
es dun medio po oso,
polo que os es udos le ados a cabo nelas poden se ep esen a i os de sis emas moi os m´
ais
complexos e de pa icula in e ese na indus ia.
O en´
omeno de dixi aci´
on en c´
elulas de Hele-Shaw oi e segue sendo cump idamen e
es udado e ca ac e izado desde hai d´
ecadas. Os p imei os es udos abo da on a p oblem´
a ica
de desp azamen os non eac i os an o miscibles como inmiscibles, demos ando a impo ancia
de ac o es como a ensi´
on supe icial ou a di usi´
on na deses abilizaci´
on dos sis emas
[93, 48, 127, 197, 8, 45].
Mais ecen emen e, es ud´
a onse os p ocesos mediados po eacci´
ons qu´
ımicas, onde
sis emas con mobilidades iniciais a o ables (e po an o, es ables), son dees abilizados po
p ocesos isicoqu´
ımicos axus ados ao desp azamen o [45, 25, 44, 9].
O es udo de eacci´
ons qu´
ımicas axus adas ao en´
omeno de dixi aci´
on oi inc emen ando
a s´
ua complexidade co empo. As´
ı, eacci´
ons elemen ais como p ocesos de neu alizaci´
on ou
de p ecipi aci´
on qu´
ımica, o on e son hoxe en d´
ıa cump idamen e ca ac e izados mos ando un
ico abanico de compo amen os. O e ec o des as eacci´
ons nas ines abilidades es ´
a mediado
p incipalmen e polo cambio nas az´
ons de mobilidade debido ao inc emen o de ac o es como
a densidade ou a iscosidade dos lu´
ıdos in oluc ados ou ´
a pe meabilidade do medio [141, 138,
137, 139, 73, 65, 9, 176].
A complexidade dos p ocesos qu´
ımicos axus ados ao desen ol emen o de ines abilidades
pode inc emen a se conside ando o que denominamos eacci´
ons qu´
ımicas complexas. Nun
con ex o xe al, es as eacci´
ons son de inidas como p ocesos qu´
ımicos que se p oducen en a ias
e apas (polo menos d´
uas) e con an con polo menos un in e media io de eacci´
on [116]. Den o
des a de inici´
on, engl´
obase un campo moi ex enso de eacci´
ons. Algunhas des as, xa o on
axus adas a ines abilidades con an e io idade, po exemplo os p ocesos au oca al´
ı icos [101, 49,
96].
No con ex o des e aballo, u iliza emos eacci´
ons qu´
ımicas cuxos mecanismos achegan un
paso m´
ais de complexidade espec o a as eacci´
ons au oca al´
ı icas e onde os in e media ios de
eacci´
on desempe˜
nan un papel undamen al pa a o desen ol emen o da din´
amica das mesmas.
As´
ı pois, nes e aballo, analiza ase o acoplamen o das ines abilidades hid odin´
amicas con dous
ipos de eacci´
ons complexas: A eacci´
on oscilan e de Belouso - Zhabo insky (BZ) [18] e as
eacci´
ons de cambio de pH [106, 108, 107, 159, 160, 145].
A eacci´
on BZ, ´
e un dos oscilado es qu´
ımicos m´
ais cump idamen e co˜
necidos. ´
E un
p oceso edox onde os in e media ios de eacci´
on oscilan no empo median e a din´
amica de
ac i ado /inhibido [82]. Es as oscilaci´
ons ap ´
ecianse con cambios de co pe i´
odicos no caso
dun sis ema pe ec amen e axi ado ou coa o maci´
on de pa ´
ons espacio empo ales en sis emas
xxii
DAR´
IO MART´
IN ESCALA VODOPIVEC
espacialmen e es endidos. Exis en a iaci´
ons des a eacci´
on. Nes e aballo en pa icula
op ouse po usa unha eacci´
on co˜
necida como eacci´
on BZ- CHD, onde o subs a o o g´
anico
da ecei a o ixinal subs i ´
uese polo compo˜
nen e CHD (1,4-Ciclohexanodiona) [113, 112, 187,
188, 190, 114]. Es a o mulaci´
on p esen a a ias an axes espec o a a ecei a o ixinal, pe o
a m´
ais signi ica i a ´
e que non p oduce di´
oxido de ca bono como p odu o secunda io, o cal ´
e
undamen al se se p e ende axus a es a eacci´
on a en´
omenos hid odin´
amicos.
Po ou a banda, as eacci´
ons de cambio de pH son eacci´
ons onde o pH do sis ema
cambia dun es ado alcalino (ou aceda) a un es ado acedo (ou alcalino) debido ´
a in e acci´
on
dos in e media ios de eacci´
on [107, 108, 159, 153, 145]. Exis en eacci´
ons de cambio de pH
ino g´
anicas, as cales son am´
en p ocesos edox e poden mos a oscilaci´
ons an o en sis emas
abe os como en sis emas semi abe os. Es es sis emas o on cump idamen e ca ac e izados
habendo unha g an lis a de o mulaci´
ons dispo˜
nibles [145]. Con odo, debido ´
a s´
ua na u eza
edox, es es sis emas non poden axus a se a elemen os o g´
anicos, como os pol´
ıme os ou
elemen os de anspo e de ´
a macos (ou d ug deli e y polo seu nome de ingl´
es).
Mais ecen emen e, desen ol ´
e onse eacci´
ons de cambio de pH o g´
anicas que a di e enza
das an e io es non son p ocesos edox, e po an o son menos ag esi as con elemen os o g´
anicos
como os an es indicados [106, 108, 107]. Es as eacci´
ons am´
en mos an compo amen os
oscila o ios en sis emas abe os e o on ela i amen e ca ac e izadas. Con odo, debido a non
se eacci´
ons do ipo edox, hai poucas o mulaci´
ons dispo˜
nibles.
Nes e aballo u iliza anse d´
uas eacci´
ons de cambio de pH o g´
anicas,
a eacci´
on de Fo maldehido-Sul i o (FS) [212, 106, 213], e a eacci´
on de
Fo maldeido-Sul i o-Gluconolac ona (FSG) [108, 107]. A p imei a ´
e am´
en co˜
necida
como eacci´
on Clock, po que o pH da soluci´
on cambia nun momen o de e minado de acedo
a alcalino. O empo de cambio depende das concen aci´
ons das especies in oluc adas. A
segunda eacci´
on, ´
e unha de i aci´
on da p imei a onde se axus a a hid ´
olise da gluconolac ona
ao cambio de pH da eacci´
on de Fo maldehido-Sul i o. A hid ´
olise con e e a gluconolac ona
en ´
acido gluc´
onico p oducindo un inc emen o pun ual do pH.
As´
ı pois, o obxec i o de es udo des a ese ´
e axus a en e se odos os en´
omenos desc i os
an e io men e. De manei a espec´
ı ica es uda anse es sis emas axus ados:
•Sis ema 1: Acoplamen o en e a eacci´
on BZ-CHD e unha ines abilidade de dixi aci´
on
inducida po lo abilidade en nunha c´
elula o ien ada e icalmen e.
•Sis ema 2: Acoplamen o en e a eacci´
on FS e unha ines abilidade de dixi aci´
on iscosa
nunha c´
elula o ien ada ho izon almen e.
•Sis ema 3: Acoplamen o en e a eacci´
on FSG e unha ines abilidade de dixi aci´
on iscosa
nunha c´
elula o ien ada ho izon almen e.
Des a manei a p e ´
endese cons u´
ı es es sis emas expe imen almen e, e aballa con eles
nun ango de pa ´
ame os onde os empos ca ac e ´
ıs icos sexan compa ables e oco an sine xias
en e ambos en´
omenos. Realiza ase a comple a ca ac e izaci´
on, expo anse hip´
o ese espec o a
os mecanismos de in e acci´
on in oluc ados e es ablece anse modelos ma em´
a icos que pe mi an
simula a din´
amica dos mesmos.
xxiii
Resumo
Es u u a da Tese
Es a aballo es ´
a di idido en es pa es. Cada pa e con a cunha in oduci´
on, un cap´
ı ulo
de m´
e odos expe imen ais e num´
e icos, un cap´
ı ulo de esul ados expe imen ais, un cap´
ı ulo de
esul ados num´
e icos e unha discusi´
on xe al. Independen emen e do ´
ıo condu o com´
un que
pos´
uen as pa es, elixiuse o ganiza o aballo des a o ma non s´
o pa a acili a a s´
ua lec u a,
sen´
on ademais po que cada pa e en si pos´
ue su icien e con ido independen e que xus i ica o
uso do o ma o.
As´
ı, os Cap´
ı ulos 2,6e10 son cap´
ı ulos que desen ol en an o o m´
e odo expe imen al
como o num´
e ico das Pa es I,II eIII espec i amen e. Na secci´
on expe imen al des es
cap´
ı ulos a opa ´
as odas as ecei as, o mulaci´
ons qu´
ımicas e p o ocolos emp egados nos
expe imen os, as´
ı como a desc ici´
on dos a anxos expe imen ais e es udos p e ios necesa ios
pa a o desen ol emen o do aballo. Na secci´
on num´
e ica incl´
uense conside aci´
ons xe ais,
como a desc ici´
on dos dominios compu acionais, a con igu aci´
on dos p og amas emp egados
nas simulaci´
ons, pa ´
ame os emp egados, en e ou os. Non obs an e, e pa a acili a a
lec u a da ese, o desen ol emen o das ecuaci´
ons undamen ais dos modelos es´
e ase pa a os
co esponden es cap´
ı ulos de esul ados num´
e icos, xa que na maio ´
ıa dos casos, es es dependen
dos esul ados expe imen ais.
Os concep os b´
asicos son in oducidos no Cap´
ı ulo 1. Nes e cap´
ı ulo expl´
ıcanse unha
se ie de cues i´
ons undamen ais pa a o en endemen o do aballo. As´
ı, desc ´
ıbense as eacci´
ons
oscilan es cl´
asicas, como a eacci´
on de Belouso - Zhabo insky e os oscilado es de pH an o
ino g´
anicos como os o g´
anicos. Tam´
en se in oducen as ines abilidades hid odin´
amicas en
c´
elula Hele-Shaw e os en´
omenos de dixi aci´
on po di e enza de iscosidade e de densidade.
Complemen a iamen e, incl´
uese unha secci´
on pa a desc ibi as p opiedades m´
ais impo an es
do ´
acido poliac ´
ılico, que ´
e un compo˜
nen e amplamen e u ilizado no desen ol emen o des e
aballo, m´
ais p ecisamen e nas Pa es II eIII. Finalmen e, desc ´
ıbese a ´
ecnica ´
op ica de
Schlie en. Es a ´
e unha ´
ecnica de isualizaci´
on de lu´
ıdos amplamen e u ilizada du an e odo a
ese e que oi undamen al pa a o en endemen o dos mecanismos ´
ısicos e qu´
ımicos obse ados
expe imen almen e.
APa e I comeza inmedia amen e despois do cap´
ı ulo in odu o io. Nes a pa e es ´
udase
o axus e en e a eacci´
on BZ-CHD, e a ines abilidade hid odin´
amica de dixi aci´
on po inducida
pola lo abilidade nunha c´
elula Hele-Shaw o ien ada e icalmen e.
No Cap´
ı ulo 3 p es´
en anse os esul ados expe imen ais ob idos pa a es e sis ema. Os
mesmos son analizados de o ma an o desc i i a como cuan i a i a, es udando os e ec os
p oducidos polos cambios na densidade (∆ρ) e da exci abilidade (ε). Es as pa ´
ame os o on
modi icados an o de manei a independen e como axus ada, e o seu e ec o oi ca ac e izado
median e o es udo de obse ables espec´
ı icos como os empos de induci´
on qu´
ımicos ( ind−C) e
hid odin´
amicos ( ind−H), a lonxi ude de onda (λH) e o pe ´
ıodo (TC). Des a o ma, p es´
en ase
un es udo de ca ac e izaci´
on comple o dos en´
omenos obse ados, pod´
endose es ablece as
p incipais in luencias da hid odin´
amica e a qu´
ımica na din´
amica do sis ema.
No Cap´
ı ulo 4 p es´
en ase un es udo qu´
ımico de allado ealizado coa inalidade de descub i
o mecanismo de ´
as dos esul ados expe imen ais no Cap´
ı ulo 3. De andi o es udo le ou
a cabo median e unha ex ensi a se ie de expe imen os de con ol e ca ac e izaci´
on qu´
ımica,
onde se u iliza on di e sas e amen as de an´
alises ales como espec o o ome ´
ıa e esonancia
magn´
e ica nuclea . Median e es es es udos, ´
e iden i icado o ac o p incipal no acoplamen o
en e a eacci´
on BZ-CHD e a ines abilidade hid odin´
amica.
xxi
DAR´
IO MART´
IN ESCALA VODOPIVEC
No Cap´
ı ulo 5, apl´
ıcanse os esul ados ob idos no Cap´
ı ulo 4 en combinaci´
on
coa os modelos cin´
e ico xa exis en es, pa a ep oduci de o ma na u al os esul ados
expe imen ais median e simulaci´
ons. As´
ı, nes e cap´
ı ulo comp´
a anse p imei amen e os
esul ados expe imen ais e os modelos cin´
e icos coa inalidade de ob e un ango de
pa ´
ame os de aballo equi alen e. Logo, p oponse un modelo num´
e ico ex ensi o de
eacci´
on-di usi´
on-con ecci´
on inclu´
ındo os mecanismos descube os e desen ol endo paso a
paso odas as ecuaci´
ons undamen ais. Finalmen e, os esul ados num´
e icos son analizados
analogamen e aos expe imen ais demos ando as equi alencias en e ´
ambolos sis emas.
APa e II comeza inmedia amen e a inaliza o Cap´
ı ulo 5. Nes e caso es ´
udase o axus e
en e unha ines abilidade hid odin´
amica de dixi aci´
on iscosa e unha eacci´
on de cambio de pH
nunha c´
elula de Hele-Shaw o ien ada ho izon almen e.
P e iamen e ao es udo do sis ema hid odin´
amico, ´
e necesa io desen ol e un sis ema
qu´
ımico onde a iscosidade sexa a ec ada po cambios no pH. Des a manei a, no Cap´
ı ulo
7desen ´
ol ese un sis ema onde se obse an expe imen almen e a iaci´
ons de empo ais
sin´
e gicas en e o pH e a iscosidade. Is o l´
og ase median e o acoplamen o en e un pol´
ıme o
sensible ao pH, (o ´
acido poliac ´
ılico), con d´
uas eacci´
ons de cambio de pH: a eacci´
on
de Fo maldehido-Sul i o e a eacci´
on de Fo maldehido-Sul i o-Gluconolac ona. ´
Ambolos
sis emas son cump idamen e ca ac e izados median e ´
ecnicas eol´
oxicas e anal´
ı icas, medindo
obse ables como os sal os de iscosidade e pH (∆pH e ∆µ espec i amen e), os cambios na
din´
amica empo al espec o a a o mulaci´
on o ixinal e os empos de induci´
on do sis ema. Os
obse ables o on es udados pa a cada especie qu´
ımica in oluc ada nas o mulaci´
ons u ilizadas.
Finalmen e, a ´
ul ima pa e des e cap´
ı ulo demos a o mecanismo de acoplamen o median e
expe imen os de con ol e o ei os al e na i as conside ando o mulaci´
ons di e sas.
No Cap´
ı ulo 8, o os desen ol emen os ob idos no cap´
ı ulo an e io son adap ados a unha
c´
elula de Hele-Shaw pa a xe a , con ´
exi o, unha ines abilidade de dixi aci´
on iscosa. Es e
sis ema es ´
udase de manei a desc i i a analizando os e ec os p oducidos polos pa ´
ame os m´
ais
impo an es como son o caudal olum´
e ico (Q) e a composici´
on das o mulaci´
ons u ilizadas.
Seguidamen e o sis ema ca ac e ´
ızase cuan i a i amen e, es udando a mo olox´
ıa dos pa ´
ons
obse ados median e o c´
alculo da ci cula idade (C). Tam´
en se u iliza a ´
ecnica de Schlie en
como un complemen o pa a dilucida o mecanismo da ines abilidade.
OCap´
ı ulo 9 p esen a os esul ados num´
e icos dos sis emas p esen ados nos Cap´
ı ulos
7e8. Na p imei a pa e, ad´
ap anse os modelos cin´
e icos exis en es das eacci´
ons de pH
a expe imen os. Is o eal´
ızase a opado en conxun o de pa ´
ame os que mello axus an os
modelos aos esul ados expe imen ais. Os modelos axus ados son analizados de manei a
an´
aloga a como se ai no Cap´
ı ulo 7. Seguidamen e, p es´
en anse as ecuaci´
ons undamen ais
dun modelo num´
e ico con ecci´
on-di usi´
on pa a simula o sis ema in oducido no Cap´
ı ulo 8.
Os esul ados son analizados medindo a ci cula idade num´
e ica da mesma o ma que se ixo
nos expe imen os. Dem´
os ase que os esul ados expe imen ais e num´
e icos conco dan m´
ais
acep ablemen e.
Seguidamen e comeza a Pa e III onde se es uda o acoplamen o en e unha eacci´
on de pH
e o en´
omeno de dixi aci´
on iscosa en c´
elula de Hele-Shaw o ien ada ho izon almen e. A´
ında
que pa ece epe ido ´
a pa e an e io , aqu´
ı da ase un paso m´
ais no inc emen o de complexidade.
Nes a pa e es ´
udanse d´
uas si uaci´
ons p o enien es dun mesmo sis ema expe imen al. Es es
casos ob ´
e˜
nense in e cambiado soamen e a soluci´
on desplazan e pola desp azada. No chamado
Caso I, anal´
ızase a xe aci´
on de pa ´
ons complexos en condici´
ons de apa en e es abilidade
hid odin´
amica. No Caso II anal´
ızase a es abilizaci´
on de dixi aci´
on iscosa median e p ocesos
xx
Lis o Figu es
8.9 Tempo al e olu ion o he ini ial condi ion in a colo less expe imen . . . . . . . 132
8.10 Finge ing onse o a (a) colo ed and a (b) colo less expe imen . . . . . . . . . . 132
8.11 Va ia ion o he no malized iscosi y µ∗...................... 133
9.1 Cha ac e iza ion o he FS-PAA model by a ying [Fo maldehyde]0. . . . . . . 136
9.2 Cha ac e iza ion o he FS-PAA model by a ying [SO32– ]0........... 136
9.3 Cha ac e iza ion o he FSG-PAA model by a ying [GLN]0........... 137
9.4 (a) Viscosi y ield o a con ol simula ion and (b) schema ics o he ini ial
iscosi y p o ile ac oss he adial coo dina e. . . . . . . . . . . . . . . . . . . . 138
9.5 Ini ial condi ion hickness as a unc ion o (a) [Fo maldehyde]0and (b) [SO32– ]0.139
9.6 Ci cula i y alues ob ained om he con ec i e simula ions as a unc ion o
[Fo maldehyde]0andQ. ............................. 140
9.7 Ci cula i y alues ob ained om he con ec i e simula ions as a unc ion o
[SO32– ]0 o Q=o 7mL/min........................... 140
9.8 Compa ison be ween nume ical iscosi y eac ion-di usion p o iles o
di e en (a) [Fo maldehyde]0and (b) [SO32– ]0.................. 142
9.9 OH–concen a ion p o iles o he ini ial condi ion (blue mixing zone) ob ained
om heRDsys em. ............................... 142
10.1 Schema ics o he con igu a ion o he displacing-displaced luid o each case
o s udy....................................... 152
10.2 Schema ics o he expe imen al se up. . . . . . . . . . . . . . . . . . . . . . . 153
10.3SEMequipmen . ................................. 155
10.4 2D-RDC domain used in he non-linea simula ions. . . . . . . . . . . . . . . . 156
10.5 Mesh independence s udy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
11.1 Compa ison be ween (a) eac i e and (b) non- eac i e cases. . . . . . . . . . . 160
11.2 Example o he pa e n dynamic obse ed a low low a e. . . . . . . . . . . . 161
11.3 E ec o he low a e on he pa e n o ma ion. . . . . . . . . . . . . . . . . . 161
11.4 Schlie en isualiza ion o an expe imen wi h Q = 3 µL/min. . . . . . . . . . . 162
11.5 Quan i a i e compa ison o he adial p o ile as a unc ion o Q. . . . . . . . . . 163
11.6 Quan i a i e compa ison o he displacing solu ion p o iles in a ela i e
imescale o (a) eac i e and (b) non- eac i e cases. . . . . . . . . . . . . . . . 164
11.7 S abiliza ion o an ini ially uns able on o (a) Q = 10 µL/min and (b) Q =
200 µL/min..................................... 165
11.8 Reac ing in e aces o se e al low a es. . . . . . . . . . . . . . . . . . . . . 166
11.9 Case II obse ed using he Schlie en echnique o (a) 5 µL/min, and (b) 500
µL/min....................................... 167
11.10Quan i a i e measu emen s o he mo phological changes o he in e ace as a
unc iono he low a eQ............................. 167
11.11Compa ison be ween he displaced olumes o he eac i e and con ol cases
o Q = 10 µL/min................................. 168
12.1 C1: Schlie en images o a con ol expe imen wi hou C·I............ 170
12.2 C2: Con ol expe imen whe e he o maldehyde was emo ed om he
displacingcomposi ion............................... 171
12.3 C3: Con ol expe imen he he displacing solu ion was composed only o PAA. 172
xxxii

DAR´
IO MART´
IN ESCALA VODOPIVEC
12.4 C4: Con ol expe imen wi h only sul i e in he displacing solu ion composi ion. 173
12.5 Equilib ium o (a) CO2and (b) SO2species in aqueous solu ion a 23°C . . . . 173
12.6 C5: Con ol expe imen whe e he displacing solu ion was composed o PAA
and CO32– . .................................... 174
12.7 P ecipi a ion es done o h ee di e en p epa a ions. . . . . . . . . . . . . . 175
12.8 SEM mic og aphs o Solu ion A and he p ecipi a e. . . . . . . . . . . . . . . . 176
12.9 Close obse a ion o he on s abiliza ion in he e e se expe imen . . . . . . 177
12.10Measu emen o he eac ion on eloci y used o es ima e he Damhk¨
ole
numbe (Da). ................................... 179
12.11Damhk¨
ole numbe (Da) es ima ion as a unc ion o he a e age in e ace adius. 180
12.12PeDa numbe es ima ion o di e en di usion a es o Solu ion B. . . . . . . . 181
13.1 Equi alence be ween he eac ion a e and he on eloci y used o he
nume ical simula ion o Case I. . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.2 Nume ical esul s o he simula ion o Case I. . . . . . . . . . . . . . . . . . . 186
13.3 E ec o changing he low a e in he nume ical simula ion o Case I. . . . . . 186
13.4 Ci cula i y calcula ion as a unc ion o he low a e o Case I. . . . . . . . . . 187
13.5 Nume ical esul s o he simula ion o Case II. . . . . . . . . . . . . . . . . . . 187
13.6 Close obse a ion o he p ecipi a ion e ec in a simula ion o Case II . . . . . 188
13.7 Simula ion esul s o he e ec o Q on Case II. . . . . . . . . . . . . . . . . . 188
13.8 Ci cula i y calcula ion as a unc ion o he low a e o Case II. . . . . . . . . . 189
13.9 Quan i a i e nume ical compa ison be ween he displaced olumes o he
eac i e and con ol cases o Q = 10 and µL/min................. 190
13.10E ec o he di usion coe icien o species B o (a) he ci cula i y and (b) he
pa e n o ma ion.................................. 191
13.11E ec o he a ia ion o K =k1/k2........................ 191
13.12Expe imen al obse a ion o he e ec o changing K . ............. 192
B.1 Capilla y eac o used in 1D eac ion-di usion expe imen s. . . . . . . . . . . 206
B.2 Schema ics o he 2D eac ion-di usion eac o . . . . . . . . . . . . . . . . . . 207
B.3 1.5D nume ical domain used o simula e he capilla y sys em. . . . . . . . . . 208
B.4 2D nume ical domain used o simula e he RD model. . . . . . . . . . . . . . . 209
B.5 Space-Time plo s ob ained om he 1D-RD capilla y sys em o di e en alues
o ε......................................... 210
B.6 Expe imen al 2D-RD expe imen s o di e en alues o ε............ 211
B.7 STPs ob ained om he non-con ec i e aga ose-based sys em. . . . . . . . . . 211
B.8 STPs ob ained om he e oin concen a ion ield by simula ing he capilla y
sys em in a 1.5D nume ical domain. . . . . . . . . . . . . . . . . . . . . . . . 213
B.9 2D-RD simula ions esul s ob ained om he modi ied skele on model o
se e alexci abili ies. ............................... 214
B.10 STPs ob ained om he 2D-RD simula ions. . . . . . . . . . . . . . . . . . . . 215
C.1 Me hod used o calcula e (TCand ind−C). .................... 218
C.2 Me hod used o calcula e λHand ind−H...................... 218
C.3 Schema ic o he p ocedu e o calcula e he ci cula i y. . . . . . . . . . . . . . 219
C.4 (a) Finge densi y a ea dAcalcula ed o Q = 20 mL/min on Figu e 8.6. Bo h
esul s ag ee wi h hose ob ained using he ci cula i y. (b) dAcalcula ed a ixed
dis ance d = 108 mm as a unc ion o he Fo maldehyde]0. ........... 220
xxxiii
Lis o Figu es
C.5 Ci cula i y calcula ion o Cases I and II . . . . . . . . . . . . . . . . . . . . . 221
C.6 P ocedu e o calcula e he a e age displacing p o iles o Case I . . . . . . . . 222
D.1 Fi ing o he solu al expansion coe icien s. . . . . . . . . . . . . . . . . . . . 224
D.2 Pe meabili y as a unc ion o he p ecipi a e concen a ion. . . . . . . . . . . . 224
D.3 2D RDC simula ion wi hou including he gene a ion o [Q·H2Q]. . . . . . . . 225
D.4 S udy o he e ec p oduced by he colo indica o in he iscosi y o he
displacingsolu ion................................. 226
D.5 Measu emen o he i s no mal s ess di e ence N1 o long-chain (PAA-4x106
gmol−1) and sho -chain (PAA-4.5x106gmol−1) polyme s. . . . . . . . . . . 228
D.6 Fi s No mal S ess Di e ence (N1) measu ed o Solu ion A and a high
elas ici y e e ence solu ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
xxxi
Lis o Tables
2.1 Base BZ-CHD ecipe used in he RDC expe imen s whe e εand ∆ρ a ied
independen ly.................................... 51
2.3 Co espondence be ween ε-[H2SO4]0and ∆ρ-[Na2SO4]0. ........... 51
2.4 Base BZ-CHD ecipe used in he RDC expe imen s whe e εand ∆ρ a ied
coupled by changing [B O3–]0and [B –]0. Bo h eagen s we e changed
simul aneously as indica ed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Co espondence be ween εand ∆ρwi h [B O3–]0. ............... 53
2.6 Recipe used o s udying changes in εdue o changes in [H2SO4]0. These
expe imen s we e pe o med in a ba ch eac o . . . . . . . . . . . . . . . . . . 54
2.7 Values o εco esponding o changes in [H2SO4]0 o he ecipe p esen ed in
TableRecipe-5. .................................. 54
2.8 Full chemical model o he BZ-CHD eac ion. . . . . . . . . . . . . . . . . . . 56
2.9 Skele on model o he BZ-CHD eac ion. . . . . . . . . . . . . . . . . . . . . 57
2.10 Pa ame e s used in he RDC simula ions. . . . . . . . . . . . . . . . . . . . . . 59
6.1 Expe imen al ecipes used o s udy he e ec o each speci ic eagen in he
dynamics o he FS-PAA eac ion. Concen a ion alues a e exp essed as w %
o he PAA and mola i y o Na2SO3and Fo maldehyde. . . . . . . . . . . . . 104
6.2 Recipe used o analyze he e ec o he [GLN]0in he FS-PAA sys em.
Concen a ion alues a e exp essed as w % o he PAA and mola i y o
Na2SO3, Fo maldehyde, and GLN. . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Solu ion con igu a ion used o s udy he in luence o changes in he displaced
luid in he Hele-Shaw cell expe imen s. . . . . . . . . . . . . . . . . . . . . . 107
6.4 Solu ion con igu a ion used o s udy he in luence o changes in he displacing
luid in he Hele-Shaw cell expe imen s. . . . . . . . . . . . . . . . . . . . . . 107
6.5 Kine ic model o he FS eac ion. . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.6 Se o ini ial condi ions used o simula e he FS-PAA sys em by a ying
[Fo maldehyde]0.................................. 109
6.7 Se o ini ial condi ions used o simula e he FS-PAA sys em by a ying [SO32– ]0.109
6.8 Kine ic model o he FSG-PAA eac ion. . . . . . . . . . . . . . . . . . . . . . 110
6.9 Se o ini ial condi ions used o simula e he FSG-PAA sys em by a ying [GLN]0.110
6.10 Se o ini ial condi ions used in he RD simula ions o s udy changes on he
hickness o he ini ial mixing egion by a ying [Fo maldehyde]0........ 112
6.11 Se o ini ial condi ions se used in he RD simula ions o s udy changes on he
hickness o he ini ial mixing egion by a ying [SO32– ]0. ........... 112
8.1 Expe imen al Damk¨
ohle numbe s......................... 134
Lis o Tables
10.1 Composi ion o Solu ions A and B. . . . . . . . . . . . . . . . . . . . . . . . . 152
10.2 Composi ion o he con ol expe imen s. . . . . . . . . . . . . . . . . . . . . . 154
10.3 Pa ame e s used in he RDC simula ions. . . . . . . . . . . . . . . . . . . . . . 157
B.1 Recipe used in he non-con ec i e aga ose 2D-Reac ion Di usion Sys em. . . . 207
xxx i
Lis o Publica ions
The ollowing lis indica es he publica ions de i ed om his esea ch ha we e used o
c ea e he hesis. I decla e ha I am he main au ho o all hese publica ions and no o he
non-doc o collabo a o was pa o hem. I also decla e ha I am p ope ly au ho ized o use
his a icles in his con ex , and hey we e no used in any o he hesis wo k. The co esponding
au ho iza ions a e de ailed in Appendix E.
(I) D. M. Escala, M. A. Bud oni, J. Ca ballido-Landei a, A. De Wi , and A. P. Mu˜
nuzu i.
Sel -o ganized a eling chemo-hyd odynamic inge s igge ed by a chemical oscilla o .
Jou nal o Physical Chemis y Le e s, 5(3):413–418, 2014.
Impac Fac o (JCR): 7.458 (2014)
Qua ile: Q1
(II) D. M. Escala, A. P. Mu˜
nuzu i, A. De Wi , and J. Ca ballido-Landei a. Tempo al iscosi y
modula ions d i en by a pH sensi i e polyme coupled o a pH-changing chemical
eac ion. Physical Chemis y Chemical Physics, 19(19):11914–11919, 2017.
Impac Fac o (JCR): 3.906 (2017)
Qua ile: Q1
(III) D. M. Escala, A. De Wi , J. Ca ballido-Landei a, and A. P. Munuzu i. Viscous Finge ing
Induced by a pH-Sensi i e Clock Reac ion. Langmui , 35(11):4182–4188, 2019.
Impac Fac o (JCR): 3.557 (2019)
Qua ile: Q1
(IV) D. M. Escala and A. P. Mu˜
nuzu i. In e ace Finge ing Ins abili y T igge ed by
a Densi y-Coupled Oscilla o y Chemical Reac ion ia P ecipi a ion. Langmui ,
35(42):13769–13781, 2019.
Impac Fac o (JCR): 3.557 (2019)
Qua ile: Q1
(V) D. M. Escala and A. P´
e ez-Mu˜
nuzu i. Cons uc ing o Decons uc ing a Fluid Ins abili y:
A Bo om-Up App oach. Submi ed, 2021.
S a us: Submi ed
The nex lis s indica e he publica ions ha a e de i ed om he hesis esea ch, bu hey
we e no included as pa o he manusc ip due o copy igh o pe mission issues:
A icles
• D. M. Escala, J. Guiu-Sou o, and A. P. Mu˜
nuzu i. Ex e nally con olled aniso opy in
pa e n- o ming eac ion-di usion sys ems. Chaos, 25(6), 2015.

Lis o Publica ions
• C. A. Middle on, C. Thomas, D. M. Escala, J. L. Tison, and A. De Wi . Imaging he
E olu ion o B ine T anspo in Expe imen ally G own Quasi- wo-dimensional Sea Ice.
In P ocedia IUTAM, olume 15, pages 95–100, 2015.
• M. A. Bud oni, L. Lemaig e, D. M. Escala, A. P. Mu˜
nuzu i, and A. De Wi . Spa ially
Localized Chemical Pa e ns a ound an A + B

Oscilla o F on . Jou nal o Physical
Chemis y A, 120(6):851–860, 2016.
Book Chap e s
• D. M. Escala, J. Guiu-Sou o, J. Ca ballido-Landei a, A. P´
e ez-Mu˜
nuzu i, and M. E.
V´
azquez-Cend´
on. Changes in buoyancy-d i en ins abili ies using a eac ion-di usion
sys em. Nume ical Me hods o Hype bolic Equa ions: Theo y and Appl., An In . Con . o
Honou P o esso E.F. To o - P oc. o he In . Con . on Nume ical Me hods o Hype bolic
Equa ions: Theo y and Appl., pages 397–400, 2013.
• J. Guiu-Sou o, D. M. Escala, J. Ca ballido-Landei a, A. P´
e ez-Mu˜
nuzu i, and
E. Ma ´
ın-O ega. Viscous inge ing ins abili ies in eac i e miscible media. Nume ical
Me hods o Hype bolic Equa ions: Theo y and Appl., An In . Con . o Honou P o esso
E.F. To o - P oc. o he In . Con . on Nume ical Me hods o Hype bolic Equa ions:
Theo y and Appl., 409:409–412, 2013.
xxx iii
Chap e 1
In oduc ion
Abs ac :This chap e will p esen he mos basics concep s ela ed o all subjec s
add essed in his wo k. In his way, a gene al and desc ip i e e iew o he mos impo an
concep s on e e y single s udied ield will be add essed. The opics will be in oduced om
simplici y o complexi y, emphasizing all ha is necessa y o unde s and he expe imen al
and heo e ical de elopmen s.
1.1 Homogeneous Sys ems and Chemical Reac ions
F om he poin o iew o an ac i e en i onmen , a homogeneous sys em is de ined as a
ma e ial sys em in which all i s in ensi e p ope ies (such as densi y, elas ici y, empe a u e,
p essu e, e c) a e cons an in he medium. In o he wo ds, i s composi ion is uni o m o e e y
poin o he sys em. Thus, a pe ec ly mixed dissolu ion o sal in wa e o a block o i on can
be conside ed homogeneous sys ems. These ypes o sys ems can be classi ied is i e di e en
ca ego ies [116, 13]:
•Dissolu ions: Sys ems composed o only one single phase. These ypes o sys ems a e
composed o a leas , one sol en and one solu e.
•Pu e subs ances: Sys ems composed o only one subs ance.
•Open sys ems: Sys ems whe e mass and ene gy a e exchanged om he medium o he
en i onmen .
•Closed sys ems: Sys ems whe e only ene gy is exchanged be ween he medium and he
en i onmen , bu no mass.
•Isola ed sys ems: Sys ems whe e nei he mass no ene gy a e exchanged wi h he
medium.
Due o i s homogeneous cha ac e , hese ypes o sys ems can be ep esen ed by a ze o
spa ial dimension model. Rega dless o he in insic complexi y o e e y indi idual case, mos
ac i e mediums can be s udied in a de e minis ic manne by using di e en ial equa ions. Thus,
he s a e and dynamics o he sys em can be desc ibed by de ining a se o ime-dependen
a iables Ci( ) = (C1( ),C2( ),...,Cn( )), and a coupled sys em o di e en ial equa ions as:
Chap e 1. In oduc ion
dCi( )
d = i(Ci( ),pi)(1.1)
whe e idesc ibes he dynamics o he sys em and depends on a se o pa ame e s
pi= (p1,p2,...,pm). The exp essions o and pdepend on he sys em o s udy [116].
The se o Eqs. (1.1) a e o en used o desc ibe he kine ics o chemical eac ions. In his
case, he dynamics a e desc ibed by he a es o eac ion o he species in ol ed.
Conside ing he ollowing homogeneous eac ion:
aA +bB +··· → eE + F +... (1.2)
whe e a,b,...,e, ,... a e he s oichiome ic coe icien s, and A,B,...,E,F,... a e he chemical
species. The a e o consump ion o con e sion a e (J) o each eac an is p opo ional o hei
co esponden s oichiome ic coe icien (i he eac ion occu s in a closed sys em), hus:
J≡ −1
a
dnA
d =−1
b
dnB
d =··· =−1
e
dnE
d =−1
dnF
d =... (1.3)
whe e nia e he moles o he species i=A,B,....
The con e sion a e, J, is an ex ensi e p ope y ha depends on he olume o he sys em.
The con e sion a e pe olume uni J/Vis de ined as he eac ion a e [116, 13]:
≡J
V=1
V−1
a
dnA
d (1.4)
is an in ensi e magni ude and depends on he empe a u e (T), he p essu e (p), and he
concen a ion o he species [116].
I he olume o he sys em emains cons an , hen:
1
V
dnA
d =dnA/V
d =d[A]
d → =−1
a
d[A]
d =... 1
e
d[E]
d =... (1.5)
whe e [i]indica es he mola concen a ion o he species i=A,B,E....
Fo mos sys ems, i was expe imen ally demons a ed ha he eac ion a e a a gi en ime
is ela ed o he concen a ion o he in ol ed species:
=k[A]α[B]β···[L]λ(1.6)
whe e kis known as he eac ion a e (o kine ic) cons an and, he exponen s α,β,...,λa e he
pa ial o de s o he eac ion. α+β+···+λ≡nis known as he o e all o de o he eac ion.
The exp ession o he eac ion a e shown in Eq. (1.6) is known as he law o mass ac ion [116].
2
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1.2 Types o Reac ions
The p esen sec ion will b ie ly in oduce se e al ypes o eac ions. All o hem a e
impo an o he de elopmen o his hesis.
Elemen a y Reac ions: An elemen a y chemical eac ion consis s o a single s ep, in
which no in e media e compounds a e obse ed. The chemical ans o ma ions occu in only
one single s ep and pass h ough a single ansi ion s a e [116, 13]. In hese eac ions, he
molecula i y and he eac ion o de s a e well de ined and can be de i ed om he s oichiome y.
The e a e se e al ypes o elemen a y eac ions, he mos common ones a e he unimolecula
( i s -o de ) and bimolecula (second-o de ) eac ions. The dynamics o hese eac ions a e
modeled by he mass ac ion law, ha assumes ha he a e o he elemen a y eac ion is di ec ly
p opo ional o he p oduc o he ac i i ies o concen a ions o he eac an s [116].
In a i s -o de elemen a y eac ion, a molecule o a chemical species dissocia es,
polyme izes, o di ec ly con e s in o one o mo e p oduc s (P). These eac ions can be
gene ally exp essed as:
AkP (R1)
whe e he eac ion a e is gi en by:
=−d[A]
d =k[A](1.7)
Some examples o unimolecula eac ions a e:
O3O2+O (R2)
H2O2(l)H2O(l)+0.5O2(g)(R3)
On he o he hand, in second-o de elemen a y eac ion, wo species A and B eac o
p oduce one o mo e p oduc s (P). These eac ions can be exp essed gene ically as ollow:
A+BkP (R4)
whose eac ion a e is gi en by:
=−d[A]
d =−d[B]
d =k[A][B](1.8)
A simple example o bimolecula eac ion can be:
C12H22O11 +H2OkC6H12O6+C6H12O6(R5)
Complex chemical eac ions: a complex chemical (o mul is ep) eac ion is a chemical
p ocess ha consis s o se e al s eps o elemen a y eac ions and has one o mo e eac ion
in e media ies. These p ocesses mus be desc ibed by eac ion mechanisms de ailing all he
in ol ed elemen a y s eps. The a e law o a complex chemical eac ion is ob ained by
combining he a e laws o he mul iple elemen a y s eps, whe e e e y elemen a y s ep ollows
3
Chap e 1. In oduc ion
was de eloped a he Uni e si y o O egon by Field and Noyes in 1974 [67]. The model was
ob ained as a di ec educ ion o he FKN mechanism and i was achie ed by he applica ion o
s anda d me hods o chemical kine ics like he a e-de e mining-s ep app oxima ions [201, 183,
130]. In con as o he B ussela o , he O egona o model includes h ee dynamic a iables ( he
inhibi o , he ac i a o , and he ca alys ), and an adjus able kine ic modula o . This makes he
model much iche in e ms o he complexi y and beha io s ha can be ep oduced by i , like
o example, bis abili y and chaos in CSTR [81, 80, 164].
The O egona o is desc ibed by he ollowing se o chemical equa ions:
A+Yk1X+P (R18)
X+Yk22P (R19)
A+Xk32X (R20)
2X k4A+P (R21)
B+Zk51
2 Y (R22)
whe e kine ic equa ions a e gi en by:
d[X]
d =k1[A][Y]+k2[X][Y]−k3[X][A]−2k4[X]2
d[Y]
d =k1[A][Y]−k2[X][Y]+ 1
2k5 [B][Z]
d[Z]
d =2k3[A][X]−k5[B][Z]
(1.19)
whe e X ep esen s he ac i a o , Y he inhibi o , and Z he ca alys which co esponds
o HB O2, B –and Fe2+ om he o iginal BZ eac ion espec i ely. is an adjus able
s oichiome ic ac o ha makes i possible o modula e he dynamics o he eac ions.
Example simula ions o bo h, he B ussela o and he O egona o kine ic models a e
p esen ed in Figu e 1.5. These simula ions we e done by using he GNU so wa e COPASI
[95].
1.4.2 The 1,4-Cyclohexanedione-B oma e-Acid Oscilla o y Reac ion
As i was men ioned, one o he seconda y p oduc s o he o iginal BZ o mula ion is CO2
(see eac ion R113). The o ma ion o ca bon dioxide can be app ecia ed as iny bubbles ha
eme ge om he BZ solu ion. In some cases, his can be de imen al o some s udies, as he
bubbles may, o example, in e ac wi h he measu emen equipmen o hinde he subsequen
analysis o he expe imen s.
In o de o o e come such an issue, in 1994, Ku in-Cs¨
o gei e al de eloped an al e na i e
o mula ion o he classical BZ eac ion by eplacing he o iginal o ganic subs a e wi h
1,4-Cyclohexanedione (he ea e CHD - Fig. 1.6) [114, 187, 188, 190, 189]. This al e na i e
o mula ion is usually called he BZ-CHD eac ion o Bubble-F ee BZ eac ion.
The eplacemen o he malonic acid no only elimina es he o ma ion o ca bon dioxide
10

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Figu e 1.5: Simula ions o he empo al dynamics o (a) he B ussela o and (b) he O egona o kine ics models.
In bo h cases [X] and [Y] ep esen he mola concen a ion o he ac i a o and he inhibi o species espec i ely.
In (b), [Z] ep esen s he mola concen a ion o he ca alys species
bu also p oduces dynamics signi ican ly di e en compa ed o hose obse ed in he o iginal
BZ sys em [123, 84, 122]. Thus, a eling shock s uc u es, long-li ed oscilla ions, ligh
sensi i i y, wa e me ging, s acked wa e on s, densely packed pa e ns, and seg ega ed clus e s
a e examples o he b oad a ie y o beha io s obse ed in he b oma e-CHD- e oin sys em
[190, 189]. All hese cha ac e is ics made he BZ-CHD o be conside ed supe io o e
he classical BZ eac ion, and mo e sui able o s udy mul i ude o di e en phenomena
[190, 188, 187, 189].
O
OH
OH
O
(a) Malonic Acid (MA)
O O
(b) 1,4-Cyclohexanedione (CHD)
Figu e 1.6: O ganic subs a es o (a) he o iginal BZ eac ion, (b) he BZ-CHD eac ion
Se e al s udies we e made o unde s and and ully cha ac e ize he mechanism o he
BZ-CHD eac ion [113, 112, 187, 188, 114, 190, 189]. An ex ensi e kine ic model was also
p oposed by Szalai e al, whe e he dynamics and p ope ies o he BZ-CHD eac ion we e
s udied o di e en ca alys s and o he unca alyzed eac ion [189, 190]. Simila o he
FKN model, educed o skele on models we e also de eloped. In such models, he main
cha ac e is ics o he sys ems a e p ese ed e en wi h a educed numbe o a iables. This
makes possible i s nume ical implemen a ion wi hou equi ing sophis ica ed compu a ional
esou ces [189, 190, 188]. The kine ic models o he BZ-CHD eac ions a e no indica ed
in his in oduc o y sec ion, as hey will be ex ensi ely used and desc ibed in he o hcoming
chap e s.
Fo his hesis, he e oin ca alyzed BZ-CHD eac ion will be p e e ed o e he classical
BZ o mula ion. This eac ion will be ex ensi ely used in Pa I o s udy i s coupling wi h
hyd odynamic sys ems.
11
Chap e 1. In oduc ion
1.5 pH-Shi ing Reac ions
A pH-shi ing eac ion is de ined as a sys em in which he hyd ogen ion plays he mos
impo an kine ic ole in he o e all dynamics. In hese sys ems, he d i ing o ce o such
changes is he a ia ion in pH, which can be as la ge as 6 pH-uni s [145, 153]. pH-shi ing
eac ions a e a well-known g oup o eac ions ha exhibi ela i ely simple chemis y. The
eac ion mechanism in hese sys ems is be e unde s ood compa ed o o he ypes o complex
eac ions [145, 153]. This is possible since he s oichiome y and he kine ics o he global
and in e media y eac ions we e ho oughly s udied, making i possible o easily iden i y he
posi i e and nega i e eedback p ocesses.
As was men ioned in Sec ion 1.3, some complex eac ions change hei dynamics
depending on he expe imen al condi ions. This is he case o mos pH-shi ing eac ions[145].
Some o mula ions exhibi oscilla ions o complex dynamics when s udied in CSTR o
semiba ch eac o s, and only a single pH swi ch when he same eac ion occu s in a ba ch
eac o . When p oducing oscilla ions, he pH-shi ing eac ions a e also known as pH-oscilla o s
[145, 153, 129].
The e a e wo main ca ego ies o pH-shi ing eac ions ha will be add essed in his
in oduc ion. One a e he so-called ino ganic, in which he key componen s a e ino ganic, and
he o he a e he so-called o ganic (o special) in which pa o i s o mula ions a e composed
o o ganic species.
1.5.1 Ino ganic pH-Oscilla o s
Ino ganic pH-oscilla o s we e well known and as ly s udied. In mos cases, s udies we e
ocused on he oscilla ing dynamics obse ed in CSTR o semiba ch eac o s. These eac ions
a e usually composed o an oxidan and a educ an species, hus hey a e edox p ocesses. A
b oad desc ip ion o all known pH-oscilla o s is p esen ed in O b´
an e al [145].
Addi ionally o he oxidan and he educ an , in some sys ems, a second subs a e is
necessa y o he oscilla ions o occu [145]. The dynamics o a pH-oscilla o will depend
on whe he a second subs a e is needed o no [145, 153]. Bo h si ua ions a e schema ized in
Figu e 1.7.
In he one-subs a e sys ems, he educ an is oxidized by a pa ial oxida ion mechanism
p oducing a eac ion in e media y. This s ep is p o on-consuming (nega i e eedback), hus i
p oduces an inc emen in he pH (S ep 1, Fig. 1.7(a). In a second s ep, he in e media y will
eac wi h bo h, oxidan and educ an h ough a o al oxida ion mechanism. In his s ep apa
om he end p oduc , p o ons a e gene a ed by an au oca aly ic mechanism dec easing he pH
(Posi i e eedback, S ep 2, Fig. 1.7(a)). Oscilla ions a e p oduced by he epe i ion o s eps 1-2.
The cycle will con inue un il he subs a es a e consumed.
In a wo-subs a e sys em, oscilla ions occu as a consequence o wo consecu i e eac ions.
In he i s s ep, p o ons a e p oduced au oca ali ically by an oxida ion mechanism causing he
pH o dec ease (posi i e eedback, S ep 1, Fig. 1.7(b)). The gene a ed p o ons a e consumed
in a second s ep in which ano he subs a e is oxidized o p oduce he end p oduc (nega i e
eedback, S ep 2, Fig 1.7(b)). As his is a p o on consuming eac ion, he pH inc eases and he
cycle s a s again. Oscilla ions will con inue un il he i s subs a e is consumed. The second
subs a e can be a educ an o a H+consuming eagen .
An expe imen al example o he dynamics o a one-subs a e oscilla o is p esen ed in
12
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Figu e 1.7(c)). These esul s we e ob ained om he B O3–/SO3– 2 sys em [145]. In his
pa icula case, a small amoun o Mn+2 was added as a ca alys . The eac ion was pe o med in
a semiba ch eac ion a 45 °C [153]. The changes in he pH we e ollowed by using a pH-me e .
Ino ganic pH-oscilla o s can be modeled by conside ing a simple skele on mechanism. This
model was sugges ed by R´
abai e al [158, 129] and i consis s on h ee s eps:
A−+H+HA (R23)
B+HA H+
H++P (R24)
C+H+Q (R25)
whe e A–is he conjuga e base o a weak acid (HA), he educ an . This species is oxidized by
B o a conjuga e base o a s ong acid (P). The second subs a e is indica ed by C and he end
p oduc is Q.
The posi i e eedback is ep esen ed by he au oca aly ic s ep in he second eac ion. The
hi d eac ion is he nega i e eedback which is he p o on consuming s ep. This simple model
can ep oduce he oscilla o y dynamics o he pH oscilla o bo h in homogeneous and spa ially
ex ended sys ems [129].
pH-Oscilla o s a e a ac i e, no only o hei simplici y bu also o hei po en ial
applica ions. One o hese applica ions is hei use o d ug deli e y in li ing sys ems. The
undamen als o such echniques will be no add essed he e, bu he idea is o couple a
pH-sensi i e ma e ial o he dynamics o he pH-shi ing eac ion o c ea e a chemo-mechanical
de ice. In his way, a speci ic d ug o compound could be speci ically eleased due o changes
in he con o ma ion o he ma e ial p oduced by dynamics in he pH [36, 174, 14, 118, 203, 5,
196, 110].
E en hough he idea is p omising, he e is a majo p oblem: Mos o he pH-sensi i e
ma e ials a e o en o ganic, and hese ino ganic pH eac ions esul oo agg essi e o such
ypes o molecules [107]. To o e come his issue, o ganic pH-shi ing eac ions we e c ea ed
Figu e 1.7: Schema ics o he mechanism o he (a) one-subs a e and (b) wo-subs a e pH-oscilla o s. (c)
Expe imen al esul s o one-subs a e pH-oscilla o . This expe imen was done in a semiba ch eac ion a 45
°C ollowing he ecipe indica ed in Po os e al o he B O3–/SO3–2-Mn+2 sys em [153]. All pH-changes we e
eco ded by using a pH-me e .
13
Chap e 1. In oduc ion
and will be discussed in he nex sec ion.
1.5.2 O ganic pH-Shi ing Reac ions
O ganic pH-shi ing eac ions a e a special ype o pH eac ions in whe e he changes
in he pH a e no p oduced by edox p ocesses bu by acid-base s eps [145]. The e a e
wo main eac ions ha will be used in his hesis: he Fo maldehyde-Sul i e (FS) and he
Fo maldehyde-Sul i e-Gluconolac one (FSG) eac ions. Bo h eac ions will be in oduced om
a comple ely desc ip i e poin o iew. As hese eac ions will be ex ensi ely used in Pa II o
he hesis, mo e in o ma ion ela ed o hei modeling and cha ac e iza ion will be add essed in
hei co esponding chap e s.
The Fo maldehyde-Sul i e Reac ion
The FS eac ion is a well-known pH-shi ing eac ion whe e he pH apidly changes om
acid o basic a e an induc ion pe iod [212, 213]. This phenomenon ecei ed he name o clock
beha io due o he ab up change in he pH. The eac ion consis s o a bu e composed o
sul i e (SO32– ) and bisul i e (HSO3–), and o maldehyde (in he o m o me hylenglycol, which
is hyd a ed o m o o maldehyde). The clock mechanism is p oduced by he consump ion
o he in e nal sul i e/bisul i e bu e by o maldehyde o p oduce hyd oxyme hanesul ona e,
which is a o maldehyde-sul i e adduc . The maximum pH ansi ion is abou 4 uni s o pH
( om 6 o app oxima ely 10). Bo h, he induc ion ime and he pH jump can be modula ed by
changing he ini ial concen a ions o he chemical species and he sul i e/bisul i e p opo ion.
The ypical clock beha io is obse ed in a ba ch eac o , howe e , oscilla ions and complex
dynamics we e obse ed in CSTR [106]. An example o he expe imen al FS clock eac ion is
p esen ed in Figu e 1.9(a). As can be seen, he pH changes a e he induc ion pe iod. Bo h, he
ini ial and he inal pH depend on he ini ial species concen a ions.
This eac ion is ex ensi ely s udied in Ko acs e al [106]. In such wo k, he sys em is
comple ely cha ac e ized and modeled, showing he changes in he dynamics p oduced by he
ype o eac o whe e he eac ion occu s.
The Gluconolac one-Fo maldehyde-Sul i e Reac ion
The FSG eac ion was c ea ed om he FS eac ion o induce delayed nega i e eedback in
he dynamics o he eac ion [108, 107]. The eac ion sha es he same composi ion as he FS
eac ion, bu including one mo e species, he D-(+)-gluconic acid δ-lac one (o gluconolac one).
The cha ac e is ic clock dynamic o he FS eac ion is coupled wi h he hyd olysis o
he gluconolac one ha p oduces gluconic acid. This p ocess is schema ized in Figu e 1.8.
The hyd olysis eac ion is base-ca alyzed, which means ha he OH–gene a ed a e he
consump ion o he sul i e/bisul i e bu e in he FS eac ion, will inc ease he a e o hyd olysis.
The coupling be ween hese wo p ocesses p oduces a single peak in he pH, which
indica es bo h posi i e and nega i e eedback. An expe imen al example is p esen ed in Figu e
1.9(b).
In CSTR his eac ion also exhibi s complex beha io s, such as oscilla ions [108, 107].
Figu e 1.10 shows an expe imen al example o he FSG eac ion conduc ed in a con inuous
s i ed ank eac o . As can be app ecia ed, he oscilla ions a e indica ed by he change in he
colo indica o (B omo hymol blue). In he acid s a e, his indica o shows a yellow colo a ion
14
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O
OH
OH
OH
O
OH
OH–
H2OHO
O
OH
OH
OH
OH
OH
Figu e 1.8: Schema ics o he hyd olysis o he gluconolac one (le ) o gluconic acid ( igh ). This p ocess is base
ca alyzed, which means ha he OH–p oduced a e he bu e consump ion in he FS eac ion inc eases he a e
o hyd olysis.
Figu e 1.9: Examples o he dynamics o (a) FS eac ion and (b) FSG eac ion in ba ch eac o . The FS eac ion
is cha ac e ized by he ab up change in he pH a e he induc ion ime p oduced by he consump ion o he
SO32– /HSO3–bu e . In he FSG eac ion, he dynamics obse ed in (a) a e coupled wi h he hyd olysis o he
gluconolac one, p o iding he delayed nega i e eedback necessa y o p oduce he single peak obse ed in a ba ch
eac o .
ha u ns in o blue in he basic s a e. The oscilla ions we e eco ded wi h a pH-me e connec ed
o a compu e . As can be seen, he oscilla ions a e o med by he cycling p ocess p oduced by
he bu e consump ion and he hyd olysis o he gluconolac one. As he subs a es/p oduc s a e
con inuously added/ emo ed o he sys em, he single peak dynamic became oscilla ing.
Figu e 1.10: Expe imen al example o he FSG eac ion in a CSTR. (a) Oscilla ions a e easily obse ed by
ollowing he changes in he colo indica o , o (b) by using a pH-me e .
15

Chap e 1. In oduc ion
1.6 Reac ion-Di usion-Con ec ion (RDC) Sys ems
The p e ious sec ions desc ibed di e en ypes o eac ions in s i ed sys ems. Howe e , i
is no always possible o homogeneously dis ibu e a mix u e. In such cases, i is impo an o
conside he e ec s p oduced by di usi e and con ec i e o ces in he ac i e medium.
Di usion mani es s in sys ems when speci ic p ope ies a e no homogeneously dis ibu ed
in he medium. Fo example, he mixing o a d ople o ink in a glass o wa e is p oduced by
he andom collisions o he ink molecules wi h he wa e molecules. The e ec o di usion in
an inhomogeneous sys em is esponsible o many complex beha io s ha a e pa o he s udy
o his hesis.
On he o he hand, many p ocesses in Na u e and he indus y occu s unde he e ec o
ad ec i e o ces, like he anspo o pollu an s in luid lows [146], con ec i e plumes in he
ocean [200, 111], dispe sion o ae osol in he a mosphe e, mixing p ocesses in eac o s [131],
e c. Mo e speci ically, he anspo o a chemical species in a eac i e medium combined wi h
di usi i y can be esponsible o a mul i ude o di e en complex phenomena [179].
A ma hema ical desc ip ion o he anspo o a chemical species can be de i ed om he
gene al con inui y equa ion as [179]:
∂Ci
∂ +∇·~
J o =Ri(1.20)
whe e ~ = (x,y,z)is he posi ion ec o , Ci=Ci(~ , )a e he concen a ions o he species i
in a speci ic posi ion and ime, Ria e he ne olume ic sou ce o each he species i, and
~
J o =~
Jdi +~
Jad is he o al lux p oduced by he di usi e (~
Jdi ) and he ad ec i e (~
Jcon ) lux.
The exp ession 1.20 ela es he a es o change o e e y species i o he low and di usion
in o and ou o a di e en ial con ol olume and conside s he gene a ion o consump ion inside
he con ol olume [179]. The di usi e lux is p oduced by he di usion o he molecules
mo ing andomly h ough he medium, and can be desc ibed by Fick’s i s law:
~
Jdi =−Di∇Ci(1.21)
whe e Di=Di(~ , )a e he di usion coe icien o each species i.
On he o he hand, he ad ec i e lux is associa ed wi h he low con ec ion by he ollowing
exp ession:
~
Jad =~uCi(1.22)
whe e ~u=~u(~ , )is he eloci y ield.
The combina ion o Eqs. (1.21)-(1.22) wi h Eq.(1.20), gi es he gene al anspo equa ion
o each chemical species:
∂Ci
∂ +∇·(−Di∇Ci+~uCi) = Ri(1.23)
Conside ing he eac i i y be ween he chemical species, he sou ce and sink e ms Ria e
now dependen on he concen a ion o he species, he e o e Ri= i(Ci, ), whe e ia e he ne
16
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eac ion a es o he chemical sys em. By combining Equa ion (1.1), and conside ing ha he
di usion coe icien s do no depend on he posi ion and ime, Equa ion (1.23) is educed o:
∂Ci
∂ +~u·∇Ci=Di∇2Ci+ i(Ci, )(1.24)
which is known as he eac ion-di usion-con ec ion equa ion o he species i.
In a non- eac i e sys em, he sou ce and sink e m can be neglec ed ob aining:
∂Ci
∂ +~u·∇Ci=Di∇2Ci(1.25)
This equa ion is known as he di usion-con ec ion equa ion and i will be undamen al o
explain non- eac i e con ec i e p ocesses [179].
In his hesis, Eqs. (1.24) and (1.25) will be undamen al in he de elopmen o nume ical
models.
1.6.1 Reac ion-Di usion (RD) Sys ems
Reac ion-di usion sys ems a e ma hema ical models ha desc ibe how spa ially ex ended
species a e a ec ed by wo main p ocesses. One is he di usion, ha acili a es he spa ial
dis ibu ion h ough he medium. The o he one is he chemical p ocess ha modula e he
c ea ion and/o consump ion be ween species. Many na u al phenomena can be modeled
by eac ion-di usion sys ems, such as somi ogenesis[149], animal pigmen a ion[147, 105],
cell di e en ia ion [38], and chemo axis [180] among o he s. Mo e speci ically, non-linea
eac ion-di usion sys ems a e in gene al, use ul o model ou -o -equilib ium phenomena
like sel -o ganiza ion and spa io- empo al pa e n o ma ion o biological sys ems. These
p oblems a e mainly d i en by spon aneous symme y b eaking in inhomogeneous media
[156, 155, 20, 115]. The e a e h ee main cha ac e is ic egimes associa ed wi h RD sys ems:
exci able [40, 152, 133], bis able [40], and oscilla ing [55, 54]:
Exci able egime: In an exci able medium, a local pe u ba ion o he species p opaga es
h ough he medium. This si ua ion is ypically obse ed in ca diac and ne ous issues,
neu ons, and in some diseases like Pa kinson’s o epilepsy. In an inhomogeneous chemical
sys em, he exci able egime is mani es ed in he o m o ci cula , plana , o iangula a eling
wa es o in he o m o a spi al [152, 133, 60], which a e o en p oduced by he up u e o a
concen ic wa e. Exci able egimes a e also cha ac e ized o show, in some speci ic condi ions,
s uc u es ha a e s a iona y in space. An example o his is p esen ed in Figu e 1.11(a).
These s uc u es a e known as Tu ing pa e ns and we e ob ained om he BZ-AOT chemical
sys em3[205]. Tu ing pa e ns we e used as he s anda d model o explain he pigmen a ion in
animals [205]. These ypes o s uc u es will be no conside ed in his hesis.
Oscilla ing egime: In an oscilla ing egime, he local pe u ba ion p opaga es h ough he
medium simila o he exci able case, bu pe iodically. The e a e plen y o oscilla ing sys ems
in Na u e. Some examples can be ound in popula ion dynamics, me abolic cycles, ca diac and
3Au ho ’s No e: hese Tu ing pa e ns we e he i s o be ob ained expe imen ally a he Uni e si ´
e Lib e
de B uxelles, home o he B ussela o model. They we e ob ained by he au ho du ing his p e-doc o al s ay in
2013-2014.
17
Chap e 1. In oduc ion
ci cadian hy hms, and many mo e. In pa icula , he oscilla ing chemical eac ions in oduced
in Sec ion 1.4 a e examples o non-linea chemical oscilla ing sys ems. Figu e 1.11(b) shows
an example o eac ion-di usion pa e ns obse ed in an inhomogenous BZ-CHD eac ion.
Bis able egime: The bis able egime is cha ac e ized by he coexis ence o wo di e en
s a es con igu ing a s able spa ial concen a ion dis ibu ion. The impo ance o hese sys ems
lies in hei capaci y o s o e in o ma ion since hey ha e wo s able s a es. This was obse ed
in he mechanism used by neu ons when disc imina ing ne e impulses [167]. Bis abili y is no
common in chemical sys ems, howe e , he e a e some examples o eac ions ha exhibi such
beha io [19, 97].
Figu e 1.11: Examples o spa io- empo al s uc u es ob ained om eac ion-di usion sys ems. (a) Fo ma ion o
spa ially s a iona y Tu ing pa e ns obse ed om he BZ-AOT sys em. (b) T a eling wa es and spi als obse ed
om he BZ-CHD eac ion.
A gene al ma hema ical desc ip ion o eac ion-di usion sys ems can be de i ed om Eq.
(1.24) by neglec ing he e ms associa ed wi h he con ec ion:
∂Ci
∂ =Di∇2Ci+ i(Ci, )(1.26)
his exp ession is known as he eac ion-di usion (RD) equa ion o he species i.
Many sys ems in Na u e can be modeled in a simpli ied manne by conside ing only wo
a iables, an ac i a o (C1) and an inhibi o (C2). This is possible by he p inciple o sla ing
[83], which elimina es all a iables ha quickly con e ge o he s a iona y s a e and he e o e,
ha e li le e ec on he dynamics o he sys em. Thus, a mo e speci ic ma hema ical model
based on his p inciple is desc ibed by:
∂C1
∂ =D1∇2C1+ 1(C1,C2, )
∂C2
∂ =D2∇2C2+ 2(C1,C2, )
(1.27)
whe e he non-linea beha io o he sys em is de e mined by he exp ession o he eac ion
e ms 1and 2.
The exis ing nume ical models o he BZ eac ion in oduced in Sec ion 1.4.1, a e capable
o ep oduce he dynamics o he exci able and oscilla o y egimes. This can be achie ed by
combining Eqs. (1.27) wi h he ma hema ical models desc ibed by Eqs. (1.18) and (1.19).
18
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Figu e 1.12 shows he esul s o simula ing he B ussela o and O egona o models wi h
di usion. Spa io- empo al pa e ns simila o hose obse ed in Figu e 1.11(b) a e p esen ed
in Figu e 1.12(a). These s uc u es we e ob ained by simula ing he spa ial B ussela o model.
In his case, he sys em is unde he oscilla ing egime, and i is cha ac e ized o show concen ic
a eling wa es wi h global oscilla ions a he cen e . Figu e 1.12(b) shows an example o he
exci able spi al ob ained by simula ing he spa ial O egona o model. In his case, he e a e no
oscilla ions and only one single s uc u e p opaga es h ough he medium.
Figu e 1.12: Simula ions o (a) oscilla ing egime ob ained by nume ical simula ion o he spa ial B ussela o
model, and (b) exci able spi al ob ained by nume ical simula ion o he spa ial O egona o model.
Reac ion-di usion pa e ns can be also ob ained om pH-oscilla o s. These eac ions a e
especially sui able o p oduce s a iona y s uc u es due o hei au oca aly ic na u e. In hese
sys ems, he ole o he ac i a o is played by he H+ion, and he inhibi o will depend on he
speci ic o mula ion [145].
Simila o he BZ eac ion, pa e ns can be obse ed depending on he ela i e di usi i ies
o he inhibi o and he ac i a o . When bo h a e app oxima ely equal, a eling wa es a e
a o ed. Tu ing-like spa io- empo al s uc u es can be ob ained by dec easing he di usi i y o
he ac i a o . This can be achie ed by in oducing la ge anions like he polyac yla e (Sec ion
1.9), which p oduces a e e sible binding e ec o such a molecule [129]. Ano he in e es ing
cha ac e is ic is ha any ob ained s uc u e can be obse ed wi h acid-base indica o s wi hou
any o he sophis ica ed me hod [186, 145].
The e a e mo e complex si ua ions on he bibliog aphy whe e eac ion-di usion sys ems
a e con olled o modi ied by ex e nal o cings like cen i ugal o ces [79, 60], ad ec ion [209,
208], elec ic ields [151], and empe a u e[30].
19
Chap e 1. In oduc ion
1
aZa
0
µ∇2~ dz =1
aZa
0
(µ∇2~u (z)−~u 00(z))dz =µ∇2~u−12µ
a2~u(1.41)
a e ob ained he so-called B inkman equa ions:
∇p≈ −12µ
a2~u+µ∇2~u+ρ~g(1.42)
B inkman equa ions a e alid o low eloci y lows in po ous and non-po ous media. Due
o he in insic cha ac e is ics o he Hele-Shaw lows, i is possible o assume ha he ic ion
e m domina es o e he iscous e m [34, 33, 93, 45]. Taking his conside a ion, Eq. (1.42) can
be educed o Da cy’s Law:
∇p≈ −12µ
a2~u+ρ~g=−µ
κ~u+ρ~g(1.43)
whe e κ=a2/12 is he in insic pe meabili y o he Hele-Shaw cell.
F om Equa ion (1.43) i is possible o obse e ha he po osi y o a Hele-Shaw cell
is a cons an φ= 1. Addi ionally, om he simpli ica ions and assump ions made on he
o iginal Na ie -S okes equa ion, i was also possible o ob ain an exp ession o he in insic
pe meabili y o he cell.
As can be deduced om Equa ion (1.43), he low inside a Hele-Shaw cell is ep esen a i e
o he low in a po ous medium. Thus, he s udy o such lows can be done using such de ices
in a con olled en i onmen . Mo eo e , an ad an age o using Hele-Shaw cells is ha , since
hey a e anspa en , hey allow a pe ec isualiza ion o luids inside he cell, which is usually
a majo limi a ion in po ous media.
1.8 Finge ing Ins abili ies in Hele-Shaw Cells
Ha ing in oduced he hyd odynamic ins abili ies and once explained he main aspec s o
he low in po ous media, Da cy’s law, and he low inside a Hele-Shaw cell, he nex sec ions
will in oduce he inge ing phenomenon in Hele-Shaw cells. This hesis aims o analyze he
coupling o hese speci ic g oups o ins abili ies and complex chemical eac ions. Howe e , i is
necessa y o b ie ly in oduce i s he main aspec s o he non- eac i e classical, and he newe
chemically-d i en ins abili ies, as a way o e iew he s a e o he a a he beginning o his
wo k. This gene al in oduc ion will add ess desc ip i ely he main gene al aspec s ela ed o
he inge ing ins abili ies, including expe imen al cases and nume ical simula ions as e e ence
examples.
1.8.1 Densi y Finge ing Ins abili y
Densi y inge ing (o Rayleigh-Taylo ) ins abili ies a e obse ed when luids o di e en
densi ies a e subjec ed o accele a ion in a di ec ion opposi e o ha o he densi y g adien
[33]. This si ua ion is o en obse ed when a dense solu ion lies on op o a ligh e one in he
g a i y ield (simila o he example p esen ed in Figu e 1.13(d)). [171, 200, 111]. A gene al
scheme o he Rayleigh-Taylo ins abili y o wo solu es A and B is p esen ed in Figu e 1.18(a).
26

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I he luids a e miscible, his ins abili y is igge ed when he densi y o he solu ion is
a ec ed by he spa ial changes in he concen a ion o he solu e dissol ed (A=A(x,y, )and
B=B(x,y, )). In his si ua ion, he densi y o he sys em is go e ned by he concen a ion o
such solu es (ρ=ρ(A,B)).
The main s abili y ac o o he sys em is he densi y di e ence be ween he wo solu ions
(∆ρ=ρA−ρB). The s abili y o he sys em inc eases p opo ionally wi h he densi y jump
be ween he uppe and lowe laye , i such a jump is a o able. This is, dense luid below a
ligh e one. The opposi e si ua ion p oduces he des abiliza ion o he luid in e ace.
A ma hema ical model o he densi y inge ing ins abili y can be ob ained by
combining he equa ion o he low inside he Hele-Shaw cell (Eq. (1.43)) wi h he
eac ion-di usion-con ec ion equa ion o solu es A and B (Eq. (1.25)). Assuming ha he low
is go e ned by Da cy’s law and no 3D e ec s occu along he cell gap, he gene al equa ions o
he p oblems a e gi en by:
∇·~u=0
∇p=−µ
κ~u+ρ(A,B)~g
∂A
∂ +~u·∇A=DA∇2A
∂B
∂ +~u·∇B=DB∇2B
(1.44)
The alidi y o his equa ion is ounded on he Boussinesq app oxima ion [22], which
assumes ha all he densi y a ia ions induced by concen a ions (and/o empe a u e) changes
a e small compa ed o he mean densi y. As a consequence, he condi ion o incomp essibili y
(∇·~u=0) o he luid is p ese ed, as he buoyancy e ec s only e ain in he ρ~g e m o he
low equa ion. The e o e, any local p essu e a ia ion p oduced by small densi y changes can
be neglec ed.
The ma hema ical model p esen ed in Eqs. 1.44 can be sol ed nume ically o ep oduce
he physical phenomenon. Figu e 1.18(b), shows he non-linea simula ion o a ypical
Rayleigh-Taylo ins abili y ob ained by CFD so wa e. As can be seen, compa ed o he
expe imen al si ua ion shown in Figu e 1.18(c), he simula ions ag ee e y well in bo h he
sys em dynamics and he shape o he inge s, alida ing a leas quali a i ely, he nume ical
model [197, 121, 45].
Un il now, i was only conside ed he uns able case, howe e non- eac i e buoyancy-d i en
ins abili ies can be ob ained e en o ini ially s able con igu a ions. In his case, he di usi i y
also plays an impo an ole in he des abiliza ion o he sys em. Figu e 1.19(a) schema izes a
si ua ion in whe e he dense luid is on he bo om o he Hele-Shaw cell, and he ligh e one is
on op. This si ua ion is ini ially s able, bu inge ing may be obse ed i di e en ial di usion
exis s, which means ha one o he solu es di uses as e compa ed o he o he one.
In he case whe e he dense luid di uses as e han he ligh e one, double-di usi e
inge ing ins abili y is ob ained (DD, [197]). An example o his ins abili y is p esen ed in
Figu es 1.13(d-d2). I can be ound in Na u e when ho saline wa e lies o e cold eshwa e o
a highe densi y. This ins abili y is esponsible o imp o e he anspo o nu ien s and/o o
con ol he empe a u e in he oceans [171, 200, 111].
This case can be modeled using he same se o equa ions used o he Rayleigh-Taylo
ins abili y, bu conside ing p ope alues o DAand DB. Some nume ical esul s o his case
27
Chap e 1. In oduc ion
Figu e 1.18: Rayleigh-Taylo Ins abili y (a) Schema ics o he ini ial luid and solu e con igu a ion. In his
ins abili y, he dense luid is on op o he ligh e one, leading o an uns able hyd odynamic ini ial condi ion. (b)
Image sequences o a Non-linea simula ion o he Rayleigh-Taylo ins abili y ob ained by nume ical in eg a ion
o Eq. 1.44. (c) Image sequences o a Rayleigh-Taylo ins abili y ob ained expe imen ally. In his pa icula case,
a dense solu ion o e oin is pu on op o doubly dis illed wa e .
a e p esen ed in Figu e 1.19(c). The d i ing o ce o his ins abili y is he local des abiliza ion
induced by he as e di usi e species [197, 51, 45].
Figu e 1.19: Densi y Finge ing Ins abili y. (a) Schema ics o he ini ial condi ion. (b) Non-linea nume ical
simula ions o he di usi e-laye con ec ion (DLC) ins abili y. (c) Non-linea nume ical simula ions o he
double-di usi e (DD) ins abili y. All simula ions we e pe o med by nume ically in eg a ing Eqs. 1.44 on CFD
so wa e.
On he o he hand, i he ligh e luid di uses as e compa ed o he dense one,
di usi e-laye con ec ion (o DLC, [197]) ins abili y is ob ained. A nume ical example o
his si ua ion is p esen ed in Figu e 1.19(b).
Wooding e al [217] demons a ed ha he e ec o he di usion in miscible luids is
analogous o he ole played by he su ace ension in he immiscible case, which is o se
28
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IN ESCALA VODOPIVEC
he ini ial leng h scale o he inge ing pa e n.
The e a e many wo ks whe e he classical buoyancy-d i en ins abili ies we e comple ely
analyzed and cha ac e ized by heo e ical and/o expe imen al pe spec i es [59, 87, 8, 9, 120,
44, 25, 197].
1.8.2 Viscous Finge ing Ins abili y
Viscous inge ing ins abili ies a e obse ed when a less iscous luid displaces a mo e
iscous one in a po ous medium. This phenomenon is also known as Sa man-Taylo ins abili y
when luids a e immiscible [93].
When luids a e miscible, he iscous inge ing ins abili y is d i en by iscosi y g adien s
which esul s om spa ial a ia ions in he concen a ion o he solu es ha go e n he iscosi y
o he solu ion such as µ=µ(A,B). This ins abili y was expe imen ally and heo e ically
s udied, o bo h, linea [168, 46, 41], and adial [93, 168, 88, 141, 44] displacemen s.
In iscous inge ing, he s abiliza ion o he sys em is go e ned by he log mobili y a io R,
which is de ined as [48, 193]:
R=lnµA
µB(1.45)
The ins abili y occu s when R>0 [44, 45]. The s abili y o he sys em dec eases p opo ionally
o he inc emen in R.
Simila o he densi y inge ing, he ma hema ical desc ip ion o he iscous inge ing
ins abili y is based on Da cy’s law bu conside ing he iscosi y changes p oduced by
concen a ions o he in ol ed solu es. This s a emen coupled wi h he anspo equa ion o
solu es A and B (Eq. (1.25)), allows o ob ain he go e ning equa ions o he non- eac i e
iscous inge ing ins abili y in a Hele-Shaw cell:
∇·~u=0
∇p=−µ(A,B)
κ~u
∂A
∂ +~u·∇C=DA∇2A
∂B
∂ +~u·∇C=DB∇2B
(1.46)
In his case, as he displacemen occu s ho izon ally and he cell gap is na ow, he buoyancy
o ces can be neglec ed [93, 45]. Thus, he co esponding e m is included in he p essu e
g adien .
Examples o expe imen al and nume ical adial iscous inge ing a e p esen ed in Figu e
1.20. In Figu e 1.20(a), a mo e iscous and colo ed solu ion o Polye hylene glycol (PEG-300)
is displaced by doubly dis illed wa e . As can be app ecia ed, he displacemen is uns able
and iscous inge ing occu s. Figu e 1.20(b) p esen s he non-linea simula ions o a iscous
inge ing ins abili y ob ained by in eg a ing nume ically Eqs.(1.46) by a CFD so wa e. As
can be seen, he shape o he nume ical inge s is e y simila o he expe imen al coun e pa ,
alida ing one mo e ime he ma hema ical model de i ed om Da cy’s law.
29
Chap e 1. In oduc ion
Figu e 1.20: Non- eac i e iscous inge ing ins abili y in a adial Hele-Shaw cell. (a) An expe imen al case
whe e a Polye hylene Glycol (PEG-300) colo ed solu ion is displaced by doubly dis illed wa e . (b) Non-linea
simula ion ob ained by in eg a ing Eqs. (1.46) and pe o med wi h CFD so wa e sui e. A Schema ic o he eagen
and iscosi ies con igu a ion is p esen ed in he second snapsho o (b).
Di usion can also igge a iscous inge ing ins abili y in an ini ially s able con igu a ion.
Mish a e al ([128]), s udied hese di usion-d i en iscous ins abili ies o a b oad a ie y o
pa ame e s. Figu e 1.21 shows he nume ical esul s o h ee di e en si ua ions. In all cases,
i is assumed ha one o he species di uses as e han he o he one (i.e. DA>DB), and he
ini ial luid con igu a ion is s able (i.e. µA>µB). In Figu e 1.21(a), pa e ns a e ob ained due
o a di e en ial di usion mechanism induced by he des abilizing e ec he slowe species.
This si ua ion is known as DNS-VF [128]. When he ins abili y is p oduced by pu ely double
di usi e mechanism simila o he buoyancy-d i en case (Fig. 1.19(c)), he ins abili y is known
as DD-VF. This si ua ion is shown in Figu e 1.21(b). Figu e 1.21(c) shows he si ua ion in which
pa e ns a e ob ained due o a di e en ial di usion mechanism induced by he des abilizing
e ec he as e species. This case is known as DNF-VF [128].
These h ee cases a e ob ained by a ying some nume ical pa ame e s associa ed wi h he
concen a ions and iscosi ies o he displacing and displaced luid[128]. The igu e is included
as an example o he e sa ili y and he b oad possibili ies ela ed o he non- eac i e iscous
inge ing and i s modeling.
30
DAR´
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IN ESCALA VODOPIVEC
Figu e 1.21: Nume ical esul s o a di usion-d i en non- eac i e iscous inge ing ins abili y o ob ained o
linea displacemen . (a) Viscous inge ing o igina ed by he des abilizing e ec o he slowe species and induced
by di e en ial di usion (DNS-VF) (b) Viscous inge ing o igina ed by pu e di usi e e ec s, simila o he double
di usion ins abili y (DD-VF) (c) Viscous inge ing o igina ed by he des abilizing e ec o he as e species
and induced by di e en ial di usion, simila o he di usi e-laye con ec ion ins abili y (DNF-VF). In all cases
DA>DBand µA>µB[128].
1.8.3 Chemically D i en Finge ing Ins abili ies
The s udy o chemo-hyd odynamic inge ing ins abili ies added a new s ep in complexi y
by coupling chemical p ocesses o he low in po ous media. The dynamics o he inge ing
phenomenon in eac i e sys ems mainly depends on he in e ac ion be ween he species
in ol ed and he low. I he species a e passi ely ad ec ed, he p ope ies o he inge ing
in e ace emain hose o he non- eac i e case p esen ed in he p e ious sec ions. This occu s
e en conside ing ha he low clea ly in luences he spa ial- empo al dis ibu ion o chemical
ins abili y.
Howe e , he e a e cases whe e he chemical eac ion plays an ac i e ole in he
de elopmen o he inge ing ins abili y, especially when hey can ac i ely in luence o igge
con ec i e mo ions when wo solu ions con aining sepa a e eagen s come in o con ac .
The key o con olling chemically d i en ins abili ies lies in he e ec on he iscosi y o
densi y p o iles p oduced by he localized gene a ion o p oduc s due o in e acial eac ion.
These changes a ec he mobili y o he luids by a o ing o educing ins abili y.
31

Chap e 1. In oduc ion
The ollowing in oduc ion will be conside simple A+B→C eac ions as he s a ing poin
o he de elopmen o mo e complex si ua ions, which is he main objec i e o his hesis. Thus,
he nex sec ions will p esen a b ie ou line o he mos common chemically induced inge ing
ins abili ies, including expe imen al and nume ical examples.
Densi y Finge ing
In addi ion o all he physics associa ed wi h he non- eac i e case, he inclusion o a new
species (C) wi h di e en densi y and di usi i y, can s ongly al e he dynamics o he sys em
in buoyancy-d i en ins abili ies. The a ia ion in he densi y p oduced by he local gene a ion
o he eac ion o he p oduc can o ce changes in he con ec ion ield o e en mixing. This
si ua ion is illus a ed in Figu e 1.22(a) o an ini ially s able case.
A ma hema ical model o his ype o sys ems can be de i ed om he classical
non- eac i e case (Eqs. (1.44)), by including he co esponding eac ion e ms in he anspo
equa ions, and he exp ession o he densi y changes p oduced by eac an s and/o p oduc s.
∇·~u=0
∇p=−µ
κ~u+ρ(A,B,C)~g
∂A
∂ +~u·∇C=DA∇2A−kAB
∂B
∂ +~u·∇C=DB∇2B−kAB
∂C
∂ +~u·∇C=DC∇2C+kAB
(1.47)
Figu e 1.22(b,c) shows nume ical examples o chemically d i en densi y inge ing in wo
hypo he ical si ua ions. Figu e 1.22(b), p esen s he case whe e C is ligh e han Aand B, hus
i loa s o he uppe pa o he eac o /domain. This si ua ion can be obse ed expe imen ally
in ins abili ies d i en by neu aliza ion eac ions [65, 9]. The second case is p esen ed in Figu e
1.22(c), whe e C is dense han Aand B. In his si ua ion, he p oduc o he eac ion sinks o
he bo om pa o he eac o /domain. This can be obse ed expe imen ally i he ins abili y is
d i en by a p ecipi a ion eac ion [44, 23, 138, 176].
The model o ρ=ρ(A,B,C)will depend on he na u e o he sys em and he chemical
species. Thus, in some si ua ions, i is use ul o conside all he species in ol ed [45], in
o he si ua ions like he one exempli ied in Figu e 1.22, only he e ec o C. In any case,
he exp ession o ρis based on he Boussinesq app oxima ion assuming a linea ela ionship
be ween he concen a ion o he species and he densi y [197, 45].
Viscous Finge ing
In he case o chemically induced iscous inge ing, he in e play be ween eac ion and
he ins abili y occu s h ough changes p oduced on he iscosi y. Thus, a chemical eac ion
can p oduce an inc emen o dec emen on he iscosi y a he miscible in e ace, a ec ing he
p ope ies o he inge ing pa e n.
The de elopmen o a model o eac i e iscous inge ing is analogous o he densi y
inge ing case. The only di e ence is he necessi y o combine he low and anspo equa ion
wi h a sui able model o he iscosi y changes due o he eac i e species. Simila o he densi y
32
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Figu e 1.22: Nume ical esul s o a densi y inge ing ins abili y induced by he eac ion A+B→C. In his case,
he densi y is assumed o be a ec ed p ima ily by he p oduc C. (a) Schema ics o he luid con igu a ion and he
spa ial loca ion o he chemical species. The sys em is ini ially s able and i is des abilized by he p oduc o he
eac ion be ween Aand B. (b) The p oduc Cis ligh e compa ed o he emaining luids p esen in he eac o .
This p oduces he inge s o ascend o he uppe pa o he eac o . (c) Same si ua ion as (b), bu Cis dense . In
his case, he inge s sink o he bo om o he eac o . All simula ions we e pe o med by using CFD so wa e.
case, he exp ession o he iscosi y unc ion depends on he na u e o he p oblem. Conside ing
hese s a emen s, a gene al desc ip ion o his p oblem is gi en by:
∇·~u=0
∇p=−µ(A,B,C)
κ~u
∂A
∂ +~u·∇C=DA∇2A−kAB
∂B
∂ +~u·∇C=DB∇2B−kAB
∂C
∂ +~u·∇C=DC∇2C+kAB
(1.48)
Figu e 1.23 shows wo examples o iscous inge ing a ec ed by a chemical eac ion o
an expe imen al case (Fig. 1.23(a)) and a nume ical case (Figu e 1.23(b)). E en hough his
ins abili y is no p oduced by he eac ion i sel , as he hyd odynamic scena io is uns able.
Howe e , his is a simple example o how a chemical eac ion can in e ac wi h a luid
displacemen . Figu e 1.23(a) shows a mo e iscous polyme ic solu ion o Poly(ac ylic acid)
and sodium bisul i e is displaced by a less iscous aqueous solu ion o o maldehyde. The
polyme ic solu ion is colo ed wi h a pH colo indica o . The eac ion locally inc eases he pH
o he sys em a he in e ace p oducing he colo indica o o change om yellow o blue. The
sys em emained uns able, bu inge s s a ed o exhibi a blue colo a ion ha p opaga es due o
he e ec o he di usion.
This si ua ion was simula ed by modeling he chemical eac ion wi h he simple A+B→C.
Resul s a e p esen ed in Figu e 1.23(b). As i is possible o see, bo h nume ical and expe imen al
dynamics p esen much simila i ies.
33
Chap e 1. In oduc ion
Figu e 1.23: Nume ical esul s o a iscous inge ing ins abili y a ec ed by he eac ion A + B →C. In his case,
he sys em is ini ially uns able. The eac i i y is e idenced by he change in he colo indica o a he in e ace
be ween luids A and B. (a) Expe imen al case whe e a colo ed polyme ic solu ion o Poly(ac ylic acid) and
sul i e is displaced by o maldehyde. The con ac be ween hese wo solu ions p oduces an inc emen in he pH
ha changes he yellowish colo a ion o he indica o in o blue. (b) Simula ion o a iscous inge ing ins abili y
coupled wi h an A + B →C eac ion. E en hough he shape o he inge s is no exac ly equal o he one obse ed
in he expe imen al case, he physical phenomenon associa ed wi h he eac ion is well ep oduced.
Chemical eac ions can also induce iscous inge ing ins abili ies by changing he
pe meabili y o he po ous ma ix. This could be p oduced by p ecipi a ion [86, 176] and
dissolu ion [102] eac ions, o ins ance. In his case, once he displacing and displaced
solu ions make con ac , he p ecipi a ion occu s a he miscible in e ace. The p ecipi a e will
locally educe he pe meabili y by des abilizing he luid on and inducing iscous inge ing
[176]. This si ua ion is logical i analyzing Da cy’s law. In nume ical e ms, a dec emen in he
pe meabili y will p oduce he same e ec as he inc emen in iscosi y as he p essu e g adien
is p opo ional o he ac o µ/κ. The physical meaning, howe e , is comple ely di e en , and
he ins abili y mechanism is a mo e complex. Simila s udies we e done conside ing s udying
he e ec s o inc easing he pe meabili y [176, 23, 138, 102].
One majo applica ion o eac i e iscous inge ing is o con ol p ocesses ha a e o en
uns able, like enhanced oil eco e y [132, 207, 41, 21] o ch oma og aphy [166, 173, 32, 42,
198]. In hese cases, s udies a e conduc ed in inding sui able eac ions ha could supp ess he
ins abili y and imp o e he luid displacemen . This would be impo an in many ields o he
indus y o science.
1.9 The Poly(Ac ylic Acid)
A e y impo an pa o his hesis is o ind a coupling be ween a pH-sensi i e ma e ial
and a pH-changing chemical eac ion. In his sense, he mos impo an eac an ha will ac
as a nexus be ween hose di e en ields is he Poly(Ac ylic Acid), o he ea e , PAA. As a key
molecule o he de elopmen o his wo k, i is use ul o in oduce and summa ize he mos
impo an aspec s ela ed o physics and he chemis y o his eagen .
34
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The PAA is p obably one o he mos well-known and accessible pH- esponsi e polyme
a ailable. I is cu en ly used in many di e en ields like he ood indus y, medicine,
cosme ics, pha maceu ics indus y, and o he s [35]. Due o i s hyd ophilic cha ac e , he PAA
can abso b a housand imes i s weigh in wa e , o ming supe abso ben gels [194].
As i s name indica es, he PAA is a linea polyme o med om ac ylic acid (Figu e 1.24).
CH2CH C
O
OH CH2CH C
O
O−+ H+
Figu e 1.24: Chemical model o he ac ylic acid dissocia ion equilib ium in aqueous solu ion.
The PAA s uc u e has a ca boxylic g oup on each monome uni o e e y wo ca bon
a oms on he main chain and beha es as a polyelec oly e in wa e due o he dissocia ion o he
acid g oups [104] (Figu e 1.25).
CH2CH
COOH
!n
CH2CH
COO−
!n
+ nH+
Figu e 1.25: Chemical model o he PAA dissocia ion equilib ium in aqueous solu ion.
By de ini ion, a polyelec oly e is a polyme composed o mac omolecules in which a
subs an ial po ion o he cons i u ional uni s con ains ionizable g oups, o bo h [91]. A
concep ual scheme o wha a polyelec oly e is p esen ed in Figu e 1.26.
Figu e 1.26: Schema ic o a polyelec oly e molecule. Speci ically o he PAA, he anionic cha ac e o he
molecule is due o he dissocia ion o he ca boxylic g oups in o ca boxyla e ions.
1.9.1 S uc u e and pH-dependence
As all poly(ca boxylic acids), he PAA is a weak polyelec oly e ha dissocia es in aqueous
solu ions and i s ioniza ion equilib ium is pH-dependen [142]. This can be obse ed in Figu e
1.25. I he H+concen a ion inc eases, he pH dec eases, a o ing he non-ionized o m o
he molecule whe e he ca boxylic g oups a e dominan . On con a y, i he pH inc eases by
dec easing he H+concen a ion, he molecule dissocia es a o ing he ionized o m whe e he
ca boxyla e ions a e p edominan .
The non-ionized g oups acili a e he gene a ion o in e chain hyd ogen bonds, which
compac s he molecule s uc u e (globula o m). This compac ed s uc u e makes PAA acidic
solu ions ha e ela i ely low iscosi y i in dilu ed solu ions. In he ionized o m, he nega i e
cha ges o e e y dissocia ed ca boxylic g oup gene a e a epulsi e e ec ha elonga es he
molecule in o an ex ended s uc u e ( odlike o m). This con o ma ional change s ongly a ec s
35
Chap e 1. In oduc ion
Figu e 1.33(b) shows he example p esen ed in Figu e 1.33(a) bu obse ed h ough Schlie en
op ics. The second case p esen s a iscous inge ing ins abili y p oduced when a mo e iscous
solu ion o suc ose is displaced by a less iscous solu ion o NaCl. As can be seen, he Schlie en
echnique is no only powe ul enough o expose he iscous inge ing ins abili y be ween wo
colo less solu ions bu i is also capable o show he densi y di e ences be ween bo h liquids.
This di e ence in densi y is app ecia ed as he black s ipes obse ed inside he inge s [87].
The e a e mo e sophis ica ed a angemen s like he z- ype Schlie en se up, whe e he
schlie en objec is loca ed be ween wo pa abolic mi o s ins ead o a pai o lenses. The ligh
beams pass h ough he es a ea o an expe imen al a angemen aligned in a z-shape o m
[172]. Figu e 1.33(d) p esen s an example o a cold ai s eam obse ed h ough his echnique.
As can be app ecia ed, his echnique inc eases he sha pening and quali y o he image bu
equi es mo e sophis ica ed equipmen and mo e space disposal.
As can be seen, he images ob ained h ough he Schlie en echnique a e sha pe and mo e
de ailed han hose ob ained h ough he Shadowg aphy. Howe e , as was p e iously indica ed,
he Shadowg aph is easie and mo e cos -e ec i e o implemen .
42

PART I:
DENSITY FINGERING INSTABILITY
DRIVEN BY THE BZ-CHD
OSCILLATOR
Mo i a ion
Oscilla ing beha io s a e common in Na u e. They a e esponsible o a b oad ange o p ocesses
anging om indus ial applica ions, me abolic cycles, pe iodic modula ions o he en i onmen , human
oscilla ions, and mo e [134]. Oscilla ing chemical eac ions a e well-known examples ha a emp o
simula e ce ain beha io s. One o he mos s udied oscilla ing eac ions is he Belouzo -Zhabo insky
eac ion (BZ) [18, 55, 54] which was used as a model o desc ibe dynamic beha io s in na u al p ocesses
[199, 185, 143, 29]. In his sense, he inhe en complexi y o he BZ eac ion, in addi ion o i s as and
ich beha io , makes i ideal o he ep oduc ion o many na u al phenomena [54].
Many p ocesses in Na u e o en sha e he common p ope y o aking place in a luid medium.
Hyd odynamic ins abili ies play an impo an pa in hese si ua ions. The spec um o s uc u es whe e
luid ins abili ies play a ole is la ge ex ending om Na u e [200, 171, 111] o he indus y [184].
Mos ecen ly, hyd odynamic ins abili ies a e conside ed in se e al ields such as oil eco e y p ocesses
[93][41, 132, 44, 207], CO2seques a ion [44, 121, 120] among o he applica ions. I is no mal o
hink abou sys ems whe e chemical and hyd odynamic ins abili ies exis a he same ime and in e ac
syne gis ically [163]. The knowledge and con ol o such p ocesses could be essen ial o unde s and
mo e complex p oblems such as ma ine pollu ion [125] o chemical ga dens[31, 86, 16, 26], among
many o he s. Jus ecen ly, and because o he complexi y o he p oblem, hese ypes o coupled sys ems
we e conside ed and analyzed [44, 26, 57, 9, 88, 48].
The o hcoming chap e s will in oduce a ho ough desc ip ion o a coupled sys em consis ing o a
bubble- ee oscilla ing Belouso -Zhabo insky eac ion (BZ-CHD eac ion [114, 187, 188, 188, 189]) and
a classical in e acial hyd odynamic ins abili y in a e ically o ien ed Hele-Shaw cell. The sys em was
designed o allow wo miscible luids o in e ac a he in e ace, so he eac ion akes place only inside
he cell. The compe i ion o he wo ins abili ies will be now con olled by wo independen pa ame e s:
on he one hand, he exci abili y ha deals wi h he shape and du a ion o he oscilla ions and, on he
o he hand, he ini ial densi y g adien ac oss he in e ace. A de ailed quan i a i e analysis o he e ec
o each pa ame e will be p esen ed. The expe imen al indings will lead o p opose a modi ica ion o
he exis ing kine ic models o he BZ-CHD eac ion [189] ha is ully capable o desc ibe he p oblem
in a spa ially ex ended con igu a ion.
The esul s he e p esen ed a e based on he wo k published in Escala e al, (2014) ([57]) and Escala
e al, (2019) ([61]).
Chap e 2
Expe imen al and Nume ical Me hods
2.1 Expe imen al Me hods
2.1.1 Hele-Shaw Cell Cons uc ion
All he densi y inge ing es s we e pe o med in a e ically a anged Hele-Shaw cell [175, 57].
This cell was buil u ilizing wo ec angula pla es (20 cm x 10 cm) o sc a ch- esis an me hac yla e
(Plexiglass®) sepa a ed by a 0.25 mm poly( e a luo oe hylene) (PTFE) space as appea ed wi hin Figu e
2.1. Two injec ion holes we e se a he uppe and lowe pa o he me hac yla e pla es (ma ked wi h A
and B in Figu e 2.1(a) espec i ely). Two addi ional holes we e d illed a bo h sides and we e u ilized
as luid ou le s (ma ked wi h C and D in Figu e 2.1(a)). These wo holes we e ho izon ally aligned o
achie e a plana in e ace once he eac i e solu ions we e injec ed. A me allic ec angula ame was
pu on each side o he cell o ix he de ice, p e en undesi able ha m o he me hac yla e and ensu e a
homogeneous p essu e dis ibu ion ha gua an ees an equally spaced gap be ween he Hele-Shaw cell’s
wo pla es.
2.1.2 Injec ion P o ocol
Two solu ions con aining sepa a ed pa s o he BZ-CHD eac ion we e injec ed u ilizing a pe is al ic
pump (Gilson Minipuls 3) using silicon ubes a ached o a chemically esis an Polyp opylene connec o
(made by CPC®, [39]) o 4 mm in e nal diame e . Fo his wo k, wo connec o models (PMC2201212
and PMC230212) we e used. Those wo models o connec o s we e speci ically chosen due o hei
chemical esis ance.
The p ocedu e o ob ain a plana ini ial condi ion was pe o med in wo s eps simila o Shi e al
[175]. Fi s , Solu ion 1 was injec ed om he lowe hole (B) keeping closed he ou le s (C and D) (Figu e
2.2(a)). Once he cell was comple ely illed wi h his solu ion, he ube wi h Solu ion 2 was connec ed
o he uppe injec ion hole (A) a oiding he inclusion o bubbles. Secondly, once bo h connec o s we e
in place, ou le s C and D we e opened and bo h liquids we e injec ed using a as low a e (Figu e
2.2(b)). Once he in e ace be ween Solu ions 1 and 2 was comple ely plana , he injec ion was s opped
and he ou le s closed. This momen was conside ed he beginning o he expe imen (Figu e 2.2(c)).
The opening o he ou le s was con olled using a single chemically esis an PTFE auce . Fo all cases
s udied in he p esen wo k, he sys em was always ini ially hyd odynamically s able. Thus, he dense
luid (in his case Solu ion 1) was always injec ed om he bo om inle [57].

Chap e 2. Expe imen al and Nume ical Me hods
Figu e 2.1: Schema ics o he Hele-Shaw cell and he injec ion connec o s. (a) F on al and (b) side iews o he
designed Cell. The Holes named A and B in he scheme (a) a e he inle s o Solu ion 2 and 1 espec i ely. C
and D a e he ou le s ha a e only open in o de o achie e a plana in e ace a he cen e o he eac o . (c) The
chemically esis an Polyp opylene connec o s we e used o injec he solu ions. The chosen ma e ial was sui able
o dealing wi h he p oduc s o he BZ-CHD eac ion. The loca ion o he PTFE space is indica ed by he ed
a ows in (b).
2.1.3 Op ical A angemen
The Hele-Shaw cell was placed in an expe imen al se up desc ibed in Figu e 2.3. Two di e en
images we e eco ded o each expe imen a he same ime. The i s one was a “naked eye” iew
(he ea e chemical iew) which was use ul o obse e all kinds o chemical phenomena like spi als,
wa es, and spa io- empo al dynamics associa ed wi h he chemis y (clea ly obse ed due o he colo
changes o he ca alys ). The second one was an image ob ained h ough he Schlie en echnique
(he ea e Schlie en iew). This Technique is use ul o de ec a ia ions in he hyd odynamic ield ha
a e impossible o obse e in he chemical iew [172].
The expe imen al se up was illumina ed by using a high-powe ligh -emi ing diode (LED) il e ed
using a a iable sli o ien ed in conco dance wi h he inge s displacemen ( e ical axis). A i s
collima o lens was placed in he ligh pa hway close o he LED ligh sou ce. The collima ed ligh beam
passed h ough he Hele-Shaw cell. A 50/50 beam spli e was placed be ween he cell and a second
collima o lens. The de ia ed ligh beam di ec ly impac s a CMOS came a (PixeLink PL-B776U) which
eco ded he chemical iew. The emaining 50% o he ligh beam passed h ough a second collima o
lens. A kni e-edge cu o il e was placed a he ocal poin o he second collima o in o de o ob ain
a Schlie en iew o he cell which was eco ded by a second CMOS came a (PixeLink PL-B776U). A
g adua ed spa ial e e ence was used o ob ain quan i a i e in o ma ion.
2.1.4 Chemical Recipes and Expe imen al Designs
In he p esen wo k, se e al ecipes o he BZ-CHD we e used depending on he expe imen al
con ex . The ollowing sec ion summa izes he eac ion and p o ocols ela ed o each speci ic pa .
All solu ions p esen ed he e we e made om eagen g ade s ocks. Mo e de ails ega ding he s ock
48
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Figu e 2.2: Schema ics o he injec ion p ocedu e. In (a) he ini ially emp y Hele-Shaw cell was ini ially illed
wi h Solu ion 1 om inpu B. (b) Once illed, Solu ion 2 was injec ed om inpu A and ou pu s C and D a e
opened. (c) Bo h liquids we e injec ed un il he plana in e ace was ob ained. A ha momen , all he inpu s and
ou pu s we e closed.
Figu e 2.3: Expe imen al se up used o he con ec i e Hele-Shaw sys em. The a angemen was buil by
eco ding wo di e en images om each expe imen . A i s came a eco ded he di ec obse a ion o he
expe imen . This obse a ion was ob ained by ilming he e lec ion o he cell h ough he beam spli e loca ed
be ween he cell and he second collima o lens. The second obse a ion was ob ained by using he Schlie en
op ical echnique. The ligh beam passed h ough he cell loca ed be ween wo collima o lenses. The kni e-edge
il e was loca ed a he ocal poin o he second collima o .
49
Chap e 2. Expe imen al and Nume ical Me hods
p epa a ions a e included in Appendix Sec ion A.1.
Bo h solu ions we e he mally s abilized a 22 °C by using a he mos a ic ba h in o de o a oid
densi y a ia ions due o empe a u e changes. The maximum eco ded empe a u e a ia ion be ween
solu ions was ∆T=|TS1−TS2|=0.2 °C. All ypes o he mal a i ac s ela ed o he ligh sou ce can be
neglec ed as he LED sou ce does no p oduce signi ican empe a u e a ia ions. All he expe imen s
we e done in a oom loca ed in a con olled en i onmen , a oiding signi ican empe a u e a ia ions
du ing each expe imen .
Two main pa ame e s we e used o his pa o he wo k: he exci abili y (ε) and he densi y
di e ence be ween solu ions 1 and 2 (he ea e densi y g adien , ∆ρ).
The exci abili y is de ined as:
ε=[H2SO4]0·[B O3−]0
[CHD]0
(2.1)
his exp ession was aken om K insky [109] and Vanag e al [205]. This alue is ob ained wi h he
ini ial concen a ion o he species o he BZ-CHD eac ion.
On he o he hand, he densi y g adien is de ined as:
∆ρ=ρS1−ρS2(2.2)
his alue is calcula ed as he di e ence be ween he densi y o he solu ion loca ed a he bo om (ρS1),
and he densi y o he luid loca ed a he uppe pa o he Hele-Shaw cell (ρS2). This pa ame e was
adop ed as a simple measu e o e alua e he changes in densi y due o he chemical a ia ions. All
densi y measu emen s we e done by using a An on-Paa DMA™ 35 densi y-me e wi h an ins umen al
p ecision o 0.001 g/cm3.
Recipe o he RDC Hele-Shaw Sys em wi h an independen change o εand ∆ρ.In his
case, εand ∆ρwe e a ied by simul aneously changing [H2SO4]0in bo h solu ions, and [NaSO4]0in
Solu ion 1 espec i ely. Bo h pa ame e s we e a ied up o ou di e en alues each ob aining a o al
numbe o six een expe imen s. Each expe imen was epea ed h ee imes. A schema ic illus a ion o
he expe imen al design is p esen ed in Figu e 2.4. The ecipe and used concen a ions a e indica ed in
Table 2.1.
Figu e 2.4: Schema ics o he expe imen al design used o s udy independen changes o εand ∆ρ. The exci abili y
o he sys em was a ied by changing [H2SO4]0simul aneously in solu ions 1 and 2, and i was calcula ed as
indica ed in Eq. (2.1). The densi y g adien was a ied by changing [Na2SO4]0in Solu ion 1. This alue was
calcula ed as indica ed in Eq. (2.2) by measu ing he densi ies o solu ions 1 and 2.
50
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Recipe o Solu ion 1 Recipe o Solu ion 2
Species Concen a ion (M) Species Concen a ion (M)
H2SO40.847, 1.693, 2.540, 3.387 H2SO40.847, 1.693, 2.540, 3.387
CHD 0.291 B O3–0.142
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 ecipe used in he RDC expe imen s whe e εand ∆ρ a ied independen ly.
Tables 2.3(a,b) show he alues o εand he a e age ∆ρcalcula ed by changing [H2SO4]0and
[Na2SO4]0 espec i ely. These calcula ions we e done by using he concen a ions indica ed 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 o εand ∆ρcalcula ed om Eqs. (2.1) and (2.2) espec i ely. (a) Values o εob ained by
changing [H2SO4]0. (b) Values o ∆ρob ained by changing [Na2SO4]0. The densi y alues a e indica ed as he
a e age alue be ween h ee eplicas ± he s anda d de ia ion.
Recipe o he RDC Hele-Shaw Sys em wi h a coupled change o εand ∆ρ.This se o
expe imen s we e made by a ying he NaB O3concen a ion in Solu ion 2. Thus, his no only changed
he sys em exci abili y bu also he densi y g adien . In addi ion, i is impo an o ema k ha he
ollowing ecipe includes sodium b omide (NaB ) in i s o mula ion, he H2SO4concen a ion in each
solu ion is no equal and i does no includes Na2SO4. The addi ion o sodium sul a e was no necessa y
51
Chap e 2. Expe imen al and Nume ical Me hods
in oduced in Sec ion 1.47 o desc ibe he dynamics o a chemically induced densi y inge ing ins abili y,
we e adap ed and nume ically sol ed by he so wa e. The eac i e e ms we e aken om he Skele on
model p esen ed in he able 2.9 and we e used oge he wi h a s i chemis y sol e . The densi y
unc ion, he pe meabili y, and he kine ic laws o he skele on model we e implemen ed h ough use
de ined unc ions (UDF). The calcula ion domain (D) and he ini ial eagen condi ions a e indica ed in
he schema ic o he Figu e 2.6. A squa ed mapped mesh o 181 elemen s in he x-di ec ion and 181
elemen s in he y-di ec ion was used as he nume ical g id. The g a i y ield was se o a alue o 9.81
m2/salong he y-axis.
As will be desc ibed in u he chap e s, he base model equa ions o he RDC simula ions ha e
been modi ied om he classical o mula ion o include he chemical species in ol ed in he ins abili y
mechanism, which is he quinhyd one complex ([Q·H2Q]). E en hough he de elopmen o such
modi ica ion is ex ensi ely de ailed in Chap e 5, some in o ma ion ega ding he con igu a ion o he
RDC model mus be included in his sec ion o comply wi h he hesis s uc u e. In his sense, he
bounda y condi ions o his pa icula case we e se as ze o di usi e lux o all he species excep o
he quinhyd one, which was se as [Q·H2Q]0= 0 a he lowe bounda y o nume ical s abili y easons.
This is also indica ed in Figu e 2.6
Figu e 2.6: 2D nume ical domain used o simula e he RDC model. The mesh consis s o 181 elemen s in he
ho izon al di ec ion and 181 elemen s in he e ical di ec ion. Ze o di usi e lux bounda y condi ions a e se a
he bounda ies excep o he [Q·H2Q].
The chemical species we e spa ially loca ed simila o hei expe imen al coun e pa by he
ollowing piecewise unc ion:
Cai(x,y,0) = ([Ca]0(x,y)(ξ (x,y)+1)y<L/2
0y⩾L/2
Cbi(x,y,0) = (0y<L/2
[Cb]0(x,y)(ξ (x,y)+1)y⩾L/2
[H2SO4]0(x,y,0) = k,∀(x,y)∈D
(2.4)
58

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whe e [Ca]0a e: [CHD]0, [Fe(phen)3]2+
0, [Na2SO4]0and [Cb]0a e: [NaB O3]0and [Fe(phen)3]3+
0. (x,y)
is a no mally dis ibu ed andom unc ion wi h ampli ude ξ= 10−2. [H2SO4]0was se cons an o he
en i e domain.
The p essu e ield was calcula ed by using a second-o de upwind scheme, while he chemical
species we e calcula ed by using a i s -o de upwind scheme. The ime s ep was au oma ically con olled
by he so wa e based on an adap i e algo i hm using he i s -o de upwind disc e iza ion [11, 12]. All
simula ions we e calcula ed o a inal ime o 200 s. All he pa ame e s used o RDC simula ions a e
lis ed in Table 2.10.
Pa ame e Value Dimension Uni Re e ence/No es
[B O3–]00.142 M expe imen al
[CHD]00.291 M expe imen al
[Fe(phen)3]2+
04×−4M expe imen al
[Fe(phen)3]3+
04×−4M expe imen al
αB O3−0.114×10−3M−1expe imen al (App. D)
αCHD 0.019×10−3M−1expe imen al (App. D)
α[Fe(phen)3]2+/3+
0
0.264×10−3M−1expe imen al (App. D)
αNa2SO40.121×10−3M−1expe imen al (App. D)
αH+0.013×10−3M−1ad hoc
[Na2SO4]00.108, 0.200 M expe imen al/adjus ed
[H+]03, 15 M model
ρ01.000 g cm−3expe imen al
µ0.001 Pa.s expe imen al
P00 Pa
L 0.01 m
Di1×10−8m2s−1all species, excep o Q·H2Q
DQ·H2Q1×10−9m2s−1es ima ed
a2.5×10−4m expe imen al
κ05.208×10−9m2expe imen al: a2/12
RK0.75 expe imen al/adjus ed
RMMQ·H2Q218.2 g mol−1 e e ence [214]
ρS1.402 g cm−3 e e ence [214]
φ1 e e ence [197]
γ100 ad hoc
VT2.5×10−8m3calcula ed om L2a
k Q·H2Q5×10−2M−1s−1es ima ed
Table 2.10: Pa ame e s used in he RDC simula ions. All alues a e exp essed in he In e na ional Sys em o Uni s
(SI) excep ing concen a ions ha a e exp essed in Mola i y.
59
Chap e 3
Expe imen al Resul s
Abs ac : This chap e will show a comple e expe imen al cha ac e iza ion o he coupling
be ween he BZ-CHD and he densi y inge ing ins abili y. Resul s a e p esen ed in bo h,
desc ip i e and quan i a i e manne , showing he e ec o he mos impo an ac o s
in ol ed in he dynamics o he sys em. The conclusions ob ained om he expe imen al
obse a ions will be undamen al o he unde s anding o he coupling mechanism and
will acili a e he subsequen de elopmen o a con ec i e model.
3.1 Gene al Sys em O e iew
The cha ac e is ic beha io o he expe imen al sys em desc ibed in Sec ion 2.1 is p esen ed in
Figu e 3.1. These esul s we e ob ained o a e e ence case whe e ε= 1.653 M and ∆ρ= 0.004 g/cm3,
which co esponds o ini ial concen a ions o [Na2SO4]0= 0.155 M and [H2SO4]0= 3.387 M.
Figu e 3.1: (a) Chemical and (b) Schlie en iews o an expe imen wi h ε= 1.653 M and ∆ρ= 0.004 g/cm3.
The expe imen s a ed wi h an ini ially s able plana in e ace. Solu ion 1 was loca ed a he bo om and solu ion
2 is a he uppe pa o he image. A = 27 min, a eling wa es we e obse ed mo ing h ough he in e ace
om he le o he igh side o he eac o . A = 47 min, inge ing ins abili y was obse ed in he Schlie en
iew. The e ec o he inge s on he in e ace was isible in he chemical iew as he in e ace became de o med.
The a eling wa es we e also a ec ed by he inge onse as hey a el h ough he inge s. In he las ame
a = 73 min, a s ong eddish p ecipi a e appea s a he inge con ou (obse ed in bo h iews). (c) Shows he
spa io- empo al d i o he ep esen a i e inge ma ked wi h a ed dashed o al in panel (b) ame 47 min. In he
ame a e indica ed he e ical and ho izon al displacemen dis ances. Figu es (a) and (b) we e adap ed om [61]
Chap e 3. Expe imen al Resul s
Figu e 3.1(a) shows a di ec obse a ion o he eco ded expe imen (chemical iew) and Figu e
3.1(b) shows he same expe imen bu obse ed h ough he Schlie en echnique. Fou ames a e
p esen ed co esponding o consecu i e imes. The ini ially s able condi ion is shown a = 0 min and
was ob ained by using he me hod exposed in Figu e 2.2. Solu ion 1 (highe densi y) was si ua ed a he
bo om and solu ion 2 (lowe densi y) was si ua ed a he uppe pa o he Helle-Shaw cell.
The eac ion s a ed in he icini y o he in e ace whe e all chemicals o he BZ-CHD eac ion
we e mixed by di usi e p ocesses. In desc ip i e e ms, s a ing om = 27 min, i can be obse ed
how chemical wa es pass h ough he in e ace in he chemical iew. No speci ic pa e n was obse ed
in he wa e dynamic as hey mo e independen ly om le o igh o he opposi e depending on he
expe imen . In his pa icula case, he wa es we e mo ing om le o igh . In o he expe imen s, wa e
collision can be also obse ed a he cen e o he image. A his s age, no signi ican mo ion in he luid
was obse ed as can be obse ed in he Schlie en iew. This was expec ed due o he ini ially s able
condi ion.
A = 47 min, he beginning o he inge ing ins abili y was obse ed in he Schlie en iew. The
de o ma ion in he in e ace obse ed in he chemical iew was di ec ly p oduced by he inge onse .
The inge ing ins abili y is shown ully de eloped a = 73 min. F om he chemical iew is possible o
obse e how he ini ial in e ace mo ed om i s o iginal posi ion ising o he uppe pa o he eac o .
In addi ion, a eddish p ecipi a e ha eme ged a he in e ace be ween bo h solu ions was obse ed. This
p ecipi a e was deposi ed in he in e ace adop ing he shape o he inge s.
Ano he in e es ing phenomenon is he inge displacemen h ough bo h e ical and ho izon al
di ec ions. Commonly, inge pa e ns o igina ed om buoyancy-d i en ins abili ies displaces e ically
due o he g a i y ac ion. Howe e , i was obse ed in some cases ha inge s also mo ed in he
ho izon al di ec ion. Figu e 3.1(c) shows he d i o a sample inge along wi h he ho izon al and
e ical di ec ions. The inge la e al excu sion and e ical displacemen we e measu ed by isola ing
he mo emen o an indi idual inge ma ked wi h a ed dashed o al in Fig. 3.1(b) ame 3. This image
was ob ained by a e aging se e al empo al snapsho s in o one single ame. Bo h displacemen s a e
indica ed in millime e s.
3.2 Desc ip i e Analysis o he E ec o ∆ρand ε
To ha e a be e unde s anding o he e ec o he densi y g adien be ween he wo solu ions and he
exci abili y o he sys em, se e al expe imen s we e pe o med by a ying he exci abili y and he densi y
g adien independen ly. These esul s we e ob ained using he expe imen al design p esen ed in Table
2.3. In Figu e 3.2, an ex ended summa y be ween all he expe imen al cases is p esen ed. All ames a e
snapsho s aken 1 h a e he beginning o each expe imen . Ho izon al and e ical axes ep esen he
densi y g adien ∆ρ(Eq. (2.2)) and he exci abili y ε(Eq. (2.1)) espec i ely. Each poin in his diag am
shows wo obse a ions, he Schlie en iew and he chemical iew o each expe imen .
By analyzing he Schlie en images, i is possible o obse e an inc emen in he s abili y o he
in e ace in line wi h he inc emen in he densi y g adien . This was expec ed as he main s abili y ac o
in buoyancy-d i en ins abili ies is he densi y g adien (Sec ion 1.8). Fo lowe alues o he exci abili y
(ε= 0.413 M and 0.826 M espec i ely), he ins abili y was comple ely supp essed once he densi y
g adien eached ∆ρ= 0.009 g/cm3. Fo la ge alues o he exci abili y (ε= 1.239 M and 1.653 M), he
inc emen in he densi y g adien did no comple ely supp ess he ins abili y bu damped he ampli ude
o he inge s down.
Changes made by inc easing he sys em exci abili y p oduced ou ema kable e ec s. Fi s ly, he
in e ace go c ossed by he inge s o lowe densi y g adien s. This e ec became less signi ican
as he densi y g adien was inc eased, which is logical as he s abili y o he sys em was inc eased
p opo ionally o ∆ρ. Secondly, he chemical wa eleng h dec eased and he wa e eloci y inc eased.
62
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This is ypically a cha ac e is ic in he chemis y o he BZ-CHD oscilla o as has been demons a ed in
p e ious wo ks [190, 189]. Thi dly, he amoun o p ecipi a e inc eased wi h he exci abili y while he
inge onse imes we e educed in e sely p opo ional o he exci abili y. Finally, he inge wa eleng h
dec eased.
Figu e 3.2: Quali a i e compa ison o all he expe imen s ealized. All snapsho s we e aken a ound 1 h a e he
beginning o he expe imen . Each pai (ε,∆ρ) shows wo expe imen al obse a ions o he same expe imen a
he same ime. Bo h, he Schlie en iew (le side) and he chemical iew ( igh side) a e plo ed o each pai (ε,
∆ρ). The sys em go s abilized by inc easing ∆ρ. This s ongly a ec s he hyd odynamic by delaying he inge
onse and s abilizing he ini ial in e ace. By inc easing ε, he inge , chemical, and p ecipi a e onse imes we e
dec eased, while he hyd odynamic and chemical wa eleng h also dec eased. This Figu e was aken om Escala
e al [61].
3.2.1 Measu ing Obse ables
In con as o he desc ip i e analysis p e iously made, se e al mac oscopic obse ables we e
calcula ed o ob ain quan i a i e measu emen s o he sys em dynamics. In he p esen wo k, ou
main obse ables we e chosen, wo o hem used o chemis y cha ac e iza ion and he o he wo
o hyd odynamics cha ac e iza ion. Fo mally, he chemical oscilla ions pe iod (TC) and he chemical
eac ion induc ion ime ind−C( his is, he ime elapsed om he s a o he expe imen ill he i s wa es
a e obse ed) we e chosen o measu e he e ec o he chemis y in he whole p ocess. On he o he
63

Chap e 3. Expe imen al Resul s
hand, he inge wa eleng h, λH, and he hyd odynamic induc ion ime ind−H(conside ed as he ime
elapsed om he beginning o he expe imen ill he inge s s a de eloping) we e he obse ables used
o cha ac e ize he hyd odynamics. The obse ables we e di ec ly measu ed om he space- ime plo s
(STP) ob ained om he expe imen al obse a ion and using he me hodology indica ed in Appendix
Sec ion C.1.
Figu es 3.3-3.4 plo s he a ia ion o he mac oscopic obse ables p e iously de ined as a unc ion
o ε o each ∆ρanalyzed. All alues a e p esen ed as he a e age alue o e all he ealiza ions and e o
ba s show he s anda d de ia ion in he measu emen s. The alue o ∆ρused o each case is indica ed
in he legend in o de o acili a e g aphic comp ehension.
As can be obse ed in Figu e 3.3(a), a ia ions in he exci abili y p oduced a ema kable e ec
on he chemical induc ion ime. Mo e p ecisely, ind−Cgo sensibly educed o high alues o ε. The
same e ec occu ed o e e y ∆ρs udied. The densi y g adien did no p oduce any signi ican change
on his obse able in any case. Fu he mo e, excep o he lowe exci abili ies, he de ia ions in he
measu emen s we e ela i ely small indica ing ha his ind−Cis a pu ely chemical cha ac e is ic ha is
independen o he hyd odynamic condi ion.
Figu e 3.3: (a) Chemical induc ion ime ind−Cand (b) Chemical pe iod TCas a unc ion o εand ∆ρ. Figu e
adap ed om Escala e al [61].
On he o he hand, he a ia ion o TCas a unc ion o εand ∆ρis shown in Figu e 3.3(b). In a
simila manne o ind−C, he chemical pe iod TCwas s ongly a ec ed by changes in εbu no e ec
was obse ed by changing ∆ρ. In his case, he la ges dispe sion was obse ed o ε= 0.413 M (lowe
exci abili y case), whe e he a e age TC alue anges be ween 9.86 min and 10.81 min. The dispe sion
d ama ically educes o ε>0.413 M and he pe iod dec eases uni o mly o each ∆ρ. Fo ε= 1.653 M
he minimum a e age TC alue anges om 27 up o 31.8 s.
Rega ding he hyd odynamics, Figu e 3.4(a,b) shows he a ia ion o ind−Hand λH espec i ely. In
he i s case, he sys em esponse o a ia ions in εand ∆ρwas mo e complex compa ed o he p e ious
obse ables. Fo ∆ρ= 0.002 g/cm3and 0.004 g/cm3, he inc emen s in εdid no show any s a is ically
signi ican a ia ion conside ing he dispe sion o he measu ed alues. Howe e , he beha io o he
sys em changed ab up ly o ∆ρ= 0.009 g/cm3and 0.011 g/cm3. Fo hese wo cases, he alue
co esponding o ε= 0.413 M was no included in he igu e as no inge ing ins abili y was obse ed
du ing he expe imen al ime. Fo ε>0.413 M, ind−Hdec eased by inc easing he exci abil y in all
cases bu no signi ican a ia ions we e obse ed by changing he densi y g adien . These esul s we e
consis en wi h p e ious wo ks [197, 9] and shown how he chemis y can a ec hyd odynamics.
64
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Figu e 3.4: (a)Hyd odynamic induc ion ime ind−Hand (b) Hyd odynamic wa eleng h λHas a unc ion o εand
∆ρ. Values o ind−Hco esponding o ε= 0.413 M o ∆ρ= 0.009 and 0.011 g/cm3we e no plo ed as no
inge ing ins abili y was obse ed du ing such expe imen s. Figu e adap ed om Escala e al [61].
Finally, as can be app ecia ed in Figu e 3.4(b), no s a is ically signi ican a ia ions o λHwe e
obse ed o mos ε alues, excep o la ge densi ies, ∆ρ= 0.009 g/cm3and 0.011 g/cm3, whe e inge
wa eleng h dec eased sligh ly in e sely o he exci abili y, anging be ween 0.14 cm up o 0.26 cm.
3.3 Coupled ∆ρand εVa ia ion by Changing [B O3–]0
Un il now, only he e ec s p oduced by independen changes in he exci abili y and he densi y
g adien we e s udied showing he majo ole played by he chemis y in he sys em beha io . Howe e ,
on he basis o Equa ion (2.1), he exci abili y can be modi ied no only by changing he acid
concen a ion bu also by changing he ini ial concen a ions o b oma e and CHD.
This sec ion will be ocused on he e ec o he b oma e in he dynamics o he sys em. Howe e ,
as his species is in he uppe laye , i is no possible o add a hea y sal o he solu ion (such as Na2SO4
in he p e ious case) o ob aining independen changes in densi y and exci abili y. Thus, bo h ∆ρand ε
will be a ec ed simul aneously by changing [B O3–]0.
In his case, he hyd odynamic ield was obse ed by using he shadowg aph echnique ins ead o he
Schlie en echnique, which was desc ibed in Sec ion 1.101. Mo e in o ma ion ega ding he expe imen al
p o ocol, ecipes, and gene al de ails can be consul ed in Sec ion 2.1.4.
By analyzing he dependence o he sys em wi h he exci abili y, di e en beha io s we e ound in
conco dance o he esul s p esen ed in Sec ion 3.2. The chemical and he hyd odynamic induc ion
imes we e much la ge o lowe alues o ε(Figu e 3.5(a), compa ed wi h he middle and high
exci abili y cases. Likewise o Figu e 3.1, phenomena as chemical wa es wi h a la ge wa eleng h
and he displacemen o he in e ace we e obse ed as well. Con ec i e inge s we e obse ed abou
200 min om he beginning o he expe imen . Simila o he lowe εcase o he esul s p esen ed in
Figu e 3.2, he inge s did no mo e beyond he ini ial in e ace, and in he same way, hey g ew in he
e ical di ec ion and mo ed along he ho izon al di ec ion. Fo la ge exci abili ies (Figu e 3.5(b,c)), he
sys em esponse was also simila o hei coun e pa in Figu e 3.2. The hyd odynamic ins abili y was
1This echnique was used due o esou ce cons ain s a he momen in which he expe imen s we e conduc ed.
65
Chap e 3. Expe imen al Resul s
Figu e 3.5: Desc ip ion o sys em beha io o (a) low (ε= 0.334 M), (b) middle (ε= 0.557 M), and (c) high
exci abili ies (ε= 0.668 M). The chemical iew is indica ed in he uppe ows and shadowg aph iew a lowe
ows o each panel. The sys em e olu ion showed a compa able beha io wi h he case p esen ed in he p e ious
sec ion o di e en inc emen s in ε. The whi e dashed line indica ed in (b) ep esen s he ho izon al cu used o
build he space- ime plo s o Figu es 3.8 and 3.6. All ames we e aken a he expe imen al imes indica ed below
each panel. The esul s a e p esen ed using a alse-colo pale e o acili a e he obse a ion. Figu e adap ed om
Escala e al [57].
igge ed as soon as chemical wa es appea ed and bo h chemical and hyd odynamic s uc u es exhibi ed
66
DAR´
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analogous wa eleng h. In addi ion o he a eling wa es, spi als we e also obse ed in e ac ing one each
o he (Figu e 3.5(c)). Those spi als we e o igina ed om he inge ips showing he deg ee o in e ac ion
be ween bo h, he chemical and he hyd odynamic p ocess.
Figu e 3.6: Space- ime plo s c ea ed by o e laye ing he chemical and he hyd odynamic pa e ns o he cases
shown in Figu e 3.5. (a) ε= 0.334 M, (b) ε= 0.557 M and (c) ε= 0.668 M. Ho izon al and e ical b igh s ipes
ea u e he chemical wa es and he hyd odynamic inge s dynamics, espec i ely. As can be no ed, as he chemical
exci abili y is inc eased he con ec i e dynamics o inge s become mo e complica ed as a esul o a complex
chemical o cing. Figu e adap ed om Escala e al [57]
A supplemen a y analysis o hese p ocesses is also p esen ed in Figu e 3.6, whe e he ansi ion
om simple o complex chemo-hyd odynamic beha io s was cha ac e ized by compa ing he space- ime
plo s o he cases p esen ed in Figu e 3.5. He e he supe imposi ion o he chemical and he shadowg aph
iews make i possible o app ecia e he co ela ion be ween he chemical and hyd odynamic pa e ns
p e iously seen in Figu e 3.5. Figu e 3.6(a) desc ibes he case o ε= 0.334 M whe e a long-wa eleng h
ain o wa es mo ed om le o igh . Fo Figu e 3.6(b) and (c), he exci abili y o he sys em
was inc eased om ε= 0.557 M and 0.668 M espec i ely. Fo hese cases, chemical wa es showed
complica ed pa e ns wi h a compe i ion be ween a ge and spi al wa es coming om di e en di ec ions
which collided in he mixing zone. As a esul , he hyd odynamic pa e n, closely ollowing he chemical
dynamics, exhibi ed an in ica e beha io ha changes wi h ime.
67
Chap e 4. De ailed Chemical Analysis
in he inge onse ime. The igu e compa es he ind−Hwi h wo ex eme alues, bo h indica ed wi h
g ey dashed lines in he plo . The line on he bo om side indica es he induc ion ime o a ca alyzed
e e ence expe imen . This alue co esponds wi h he expe imen whe e no sal was added. On he uppe
side is indica ed he onse ime o a non-ca alyzed case. This alue was ob ained om an expe imen
wi hou e oin. As is possible o see, ind−Happ oached he non-ca alyzed ime o he highes NaCl
concen a ion. The in e se si ua ion was obse ed when he sal concen a ion was one o de less. The
o al inc emen in he induc ion ime was up o 620 %.
Aside om he delay in he induc ion ime and he inhibi ion o he spa io- empo al pa e ns, no
o he signi ican e ec was obse ed in he sys em dynamics.
Figu e 4.1: E ec o he addi ion o sodium chlo ide in he BZ-CHD eac ion. (a) F ames aken om he chemical
iew o NaCl = 0.0040 M and (b) NaCl = 0.005 M. (c) Hyd odynamic induc ion ime ( ind−H) a ia ion due o
he addi ion o NaCl s udied o [NaCl]0= 0.001 M, 0.002 M, 0.003 M, 0.004 M 0.005 M, 0.007 M and 0.01
M espec i ely. The emaining eagen s we e kep cons an as indica ed in Table 2.4. Fo his pa icula case,
[B O3–]0= 0.095 M which co esponds o ε= 0.557 M. The expe imen al ames we e colo ed o acili a e he
isualiza ion.
4.1.2 Expe imen s Wi hou CHD
The p e ious esul s indica ed ha inge ing ins abili y occu s wi h he unca alyzed eac ion. On
he o he hand, he p esence o he o ganic subs a e is undamen al o ob aining chemical oscilla ions
[113]. The e o e, i is necessa y o s udy he ole played by CHD in he ins abili y. Thus, a con ol
expe imen whe e he CHD was emo ed om he sys em is p esen ed in Figu e 4.2. Fo his case, he
base ecipe ha was used is p esen ed in Table 2.4. The emo al o CHD was compensa ed wi h he
addi ion o Na2SO4in Solu ion 1 o ob ain a compa able densi y jump.
Figu es 4.2(a,b) show he chemical and shadowg aph iews espec i ely. As can be seen, bo h cases
show a di usi e on in which he e oin was oxidized in o e iin. Nei he chemical no inge ing
pa e ns we e obse ed du ing he whole expe imen . This esul demons a ed ha CHD is a key species
in he ins abili y mechanism. Also, i was use ul o disca d he hypo hesis ha he ins abili y may be
gene a ed by he change in he oxida ion s a e o he ca alys [26].
74

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Figu e 4.2: (a) Chemical iew ow and (b) shadowg aph iew ow o an expe imen whe e [B O3–]0= 0.095 M
(which co esponds o ε= 0.557 M and ∆ρ= 0.002 g/cm3) whe e CHD was emo ed om he ecipe. No e ha
nei he he ins abili y no pa e n- o ma ion is obse ed. The expe imen al ames we e colo ed o acili a e he
obse a ion.
4.2 UV-Vis Spec oscopy
The sys em was s udied in a ully s i ed ba ch eac o by using UV-Vis spec oscopy echniques as
indica ed in he co esponding Sec ion 2.1.5. The ecipe used o his case is indica ed in Table 2.6. The
chemical beha io was cha ac e ized simila ly as he con ec i e sys em, whe e he empo al dynamic
and he chemical obse ables we e calcula ed. The dynamics o he sys em is p esen ed in Figu e 4.3(a)
o h ee di e en cases: ε= 0.207, 0.826, and 1.653 M. These alues we e ep esen a i es o h ee
di e en exci abili y condi ions, as low, middle, and high exci abili ies espec i ely. Fo he lowe case,
no oscilla ions we e obse ed bu only a single shi om he oxidized (Fe+3) o he educed (Fe+2) s a e
o he ca alys . Fo he middle and he high exci able cases, oscilla ions we e obse ed o a conside able
ange o ime. Those oscilla ions a e shown in Figu e 4.3(b) whe e a deep iew o he oscilla o y egion
co esponding o he high exci abili y case is p esen ed in he inse (c) o he igu e. In addi ion, a signal
sa u a ion egion was obse ed (Figu e 4.3(a)). This sa u a ion occu ed in expe imen s wi h medium
and high exci abili ies whe eas he abso bance signal diminished a e he shi in he oxida ion s a e o
ε= 0.207 M.
F om he empo al dynamics is possible o obse e he e ec o he exci abili y on he o e all
sys em beha io . By inc easing he exci abili y, he induc ion ime was educed. This was pa icula ly
no iceable o he low exci abili y case whe e he induc ion ime was much la ge compa ed wi h he
o he wo cases. Rega ding he oscilla ions, he inc emen in he exci abili y educed he o al ampli ude
o he oscilla ions, showing a p og essi e inc emen in he ampli ude in ime, whe eas o he middle
exci abili y case, he ampli ude emained cons an o all he oscilla o y egion. Also, one las oscilla ion
was obse ed o he middle and high exci abili y cases.
All hese chemical cha ac e is ics we e also quan i a i ely analyzed by measu ing he chemical
75
Chap e 4. De ailed Chemical Analysis
Figu e 4.3: (a) Dynamic cha ac e iza ion o he ba ch sys em o ε= 0.207, 0.826, and 1.653 M by using UV-Vis
spec oscopy. All spec a we e ob ained a a ixed wa eleng h o 510 nm. The abso bance alues we e con e ed
in o he mola concen a ion conside ing an abso p i e mola coe icien ξmax = 11000 L/(mol.cm)[68]. Only
h ee cases a e shown o acili a e he analysis. (b) spec um co esponding o he high exci abili y case whe e he
oscilla ions egion is shown in he igu e inse (c) o a ange o ime o 100 s. The sa u a ion egion is indica ed in
he blue dashed squa e. This Figu e was adap ed om Escala e al [61]
pe iod (TC) and induc ion ime ( ind−C). All he in o ma ion is p esen ed in Figu e 4.4 and was ob ained
by di ec measu emen o he spec oscopy spec a done o all he cases indica ed in Table 2.6. The
induc ion ime cha ac e iza ion is shown in Figu e 4.4(a). As can be seen, he induc ion ime go
signi ican ly educed con o ming he exci abili y was inc eased. I is in e es ing o app ecia e ha no
only he end o ind−Cis simila o he con ec i e case (Figu e 3.3(a), bu also he induc ion ime alues
we e app oxima ely o he same o de . This also con i med he majo ele ance o he chemis y in he
RDC case.
The chemical pe iod was also cha ac e ized and i is p esen ed in Figu e 4.4(b). Fo his case, he
pe iod o he oscilla ions was in he o de o seconds a he han minu es as obse ed in he RDC case
(Fig. 3.3(b)). Howe e , he end in he pe iod was compa able wi h he con ec i e case showing a
signi ican dec emen due o he exci abili y inc emen . Oscilla ions we e obse ed om ε⩾0.619 M.
76
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Figu e 4.4: Chemical obse ables cha ac e iza ion ob ained o a b oad ange o exci abili ies as indica ed in Table
2.7. (a) Chemical induc ion ime ( ind−C) and (b) chemical pe iod (TC), o he ba ch sys em. Fo (b) cases ε=
0.206 and 0.413 M we e no included in he plo as no oscilla ions we e obse ed.
4.3 P ecipi a e Fo ma ion
The chemical analysis done con i med he main ole played by chemis y in he con ec i e dynamic.
The chemical obse ables o he ully s i ed sys em quali a i ely conse ed he main cha ac e is ics o
he spa ially ex ended sys em. The con ol expe imen s demons a ed he s ong in luence o he CHD
in he ins abili y de elopmen and he e ec o emo ing he ca alys . Finally, simila dynamics we e
obse ed when changing he exci abili y by a ying [H2SO4]0and [B O3–]0. All hese esul s sugges ed
he possibili y ha he ins abili y was p oduced by a species de i ed om he o ganic subs a e (CHD).
F om he UV- is spec a, i was obse ed a signal sa u a ion o medium and la ge exci abili ies
(Fig. 4.3). On he o he hand, he eddish p ecipi a e obse ed in inge ing ins abili y also occu ed o
medium and la ge exci abili ies (Fig. 3.2). The e o e, hese esul s sugges ed ha he species gene a ed
in he sa u a ion egion would be ela ed o he ins abili y mechanism.
Se e al expe imen s we e conduc ed o unde s and how he signal sa u a ion was p oduced. Figu e
4.5(a) shows a se o ames o he BZ-CHD eac ion done in a s i ed assay ube which is di ec ly
compa able wi h a spec oscopic expe imen . The ini ial concen a ions used o his case we e he
same as used in he expe imen p esen ed in Figu e 4.3(b) (ε= 1.653 M), which co esponds o a high
exci abili y case. The exac momen when wo oscilla ions occu ed is indica ed be ween 849.6-852.4
s and 1216-1227 s. Ini ially, he e oin was in an oxidized s a e ( e iin) becoming in a educed s a e
om 1600 s. F om ha ime, no mo e oscilla ions we e obse ed. In an ins an be ween 2520 and 2880
s, he solu ion suddenly changed i s colo a ion becoming blackish and u bid. This co esponds wi h he
signal sa u a ion obse ed in he UV-Vis spec um (Fig. 4.3(b)). I is also in e es ing o obse e how he
colo a ion o he solu ion changed in ime, s a ing om a deep blue colo a ion ( e iin), acqui ing hen
a g eenish one, hen ed o inally ge colo ed in o black. This sugges ed he idea ha some seconda y
species o p oduc s also a ec ed he ypical colo a ion o he edox indica o .
A simila si ua ion was obse ed in he unca alyzed eac ion which is p esen ed in Figu e 4.5(b).
Fo his case, he solu ion was ini ially colo less. As ime p og esses, he solu ion acqui ed a
yellowish colo a ion. This colo a ion became s onge , u ning in o o ange a ound 2880 s. Like he
ca alyzed expe imen , he solu ion became u bid in a ime-lapse be ween 2880 and 3600 s. The
77
Chap e 4. De ailed Chemical Analysis
Figu e 4.5: Fully s i ed expe imen s o (a) ca alyzed BZ-CHD eac ion and (b) unca alyzed BZ-CHD eac ion
bo h o ε= 1.653 M. The emaining eagen s we e kep as indica ed in Table 2.6. Figu e adap ed om Escala e
al [61]
yellowish colo a ion in he unca alyzed expe imen explained why he ca alyzed case acqui ed a g eenish
colo a ion. This e ec was p oduced by he mix u e be ween he blue colo a ion o he oxidized s a e o
he e oin and he yellow colo a ion o he obse ed in he unca alyzed expe imen .
The u bid colo a ion obse ed in bo h cases was due o he eme gence o a hea y p ecipi a e
suddenly o igina ed a he end o each expe imen . In pa icula , he p ecipi a e onse showed some
delay in he unca alyzed expe imen compa ed o he ca alyzed one. This also led o he suspec ha he
a e o p oduc ion o he p ecipi a e is ca alyzed by he e oin. This p ecipi a e was also obse ed in he
con ec i e expe imen s as was exposed in Sec ion 3.1.
Addi ionally, all hese ac s we e also con i med by measu ing he colo in ensi y a ia ion as a
unc ion o ime. These alues we e aken om bo h expe imen al eco dings by s udying he empo al
p o ile o a sample egion indica ed wi h he wi he dashed line in Figu e 4.5(a). The esul s, p esen ed
in Figu e 4.6, show he simila i ies be ween he esul s ob ained om a comple ely independen me hod
and hose ob ained om spec oscopy. The oscilla ions in he ca alyzed expe imen (Fig. 4.6(a,c)) and
he p ecipi a e dynamics o he unca alyzed expe imen (Fig. 4.6(b)), we e obse ed wi h his me hod.
These esul s demons a ed, in quali a i e e ms, ha he signal sa u a ion was p oduced by he p ecipi a e
o ma ion.
Rega ding he con ec i e sys em, as p e iously shown in Figu es 3.1, 3.2, and 3.9, he p ecipi a e
was also obse ed a he end o he expe imen s. In Figu e 4.7, a de ailed obse a ion o an unca alyzed
expe imen shows he p ecipi a e o ma ion (Fig. 4.7(b)) o a middle exci abili y case (ε= 1.239 M).
In Figu e 4.7(a) he inge ing pa e n was obse ed due o he change in colo a ion p oduced by he
localized gene a ion o he p ecipi a e. In Figu e 4.7(c), i was also possible o app ecia e how he
p ecipi a e mo ed downwa ds h ough he Hele-Shaw cell. Figu e 4.7(d) shows an enla ged image o he
p ecipi a e pa icles sinking.
Finally, as explained in he In oduc ion (Sec ion 1.8.3), some buoyancy-d i en
chemo-hyd odynamic ins abili ies may be p oduced by he eme gence o hea y p oduc a he
in e ace be ween bo h liquids. I is due o his obse a ion ha he species in ol ed in he p ecipi a e
o ma ion may be esponsible o he de elopmen o he ins abili y. I is hen impo an o ully
cha ac e ize no only i s s uc u e bu he possible mechanism o gene a ion o his compound.
78
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Figu e 4.6: Dynamics o he (a) ca alyzed and (b) unca alyzed BZ-CHD eac ions ob ained by di ec measu emen
o he esul s p esen ed in Fig. 4.5. The plo s we e cons uc ed by analyzing he changes in he colo in ensi y o he
a o emen ioned expe imen s. (a) Dynamics o he ca alyzed BZ-CHD eac ion. (b) Dynamics o he unca alyzed
BZ-CHD eac ion. (c) Zoom o he oscilla o y egion showed in (a). In bo h cases, his me hod ep oduced he
sa u a ion egion obse ed h ough spec oscopic echniques.
Figu e 4.7: F ames o an unca alyzed expe imen wi h middle acid concen a ion. (a) An enhanced image o he
inge ing o ma ion ob ained by di ec obse a ion. (b, c) P ecipi a e o ma ion a e he inge onse . (d) Zoomed
image o he p ecipi a e mo ing downwa d he Hele-Shaw cell. ([CHD]0= 0.291 M, [B O3–]0= 0.142 M, and
[H2SO4]0= 2.540 M). This Figu e was aken om Escala e al [61]
79

Chap e 4. De ailed Chemical Analysis
4.4 Nuclea Magne ic Resonance (NMR) Spec oscopy
The nex s ep done o cha ac e ize he na u e o he p ecipi a e was o analyze he compound by
a mo e sophis ica ed spec oscopic ool. The nuclea magne ic esonance (NMR) was hen used o
elucida e he chemical s uc u e o he p ecipi a e [24]. The i s ask was o ex ac he p ecipi a e
om he eac ion beake . This has been done by il e ing and d ying he BZ-CHD solu ion as explained
in Sec ion 2.1.6. An image o he ex ac ed p ecipi a e is shown in Figu e 4.8.
Figu e 4.8: P ecipi a e ex ac ed om he homogeneously s i ed BZ-CHD eac ion. The p ecipi a e was ob ained
by il e ing and d ying as explained in Sec ion 2.1.6.
Fo his analysis bo h, Hyd ogen-1 (1H) and Ca bon-13 (13C) NMR we e pe o med by using wo
di e en sol en s: deu e a ed dime hyl sul oxide (DMSO-d6) and deu e a ed me hanol (Me OH-d4). The
espec i e esul s a e p esen ed in Figu es 4.9-4.10. Fo he DMSO-d6, Figu e 4.9 shows he 13C (Figu e
4.9(a)) and 1H (Figu e 4.9(b)) spec a. The measu ed chemical shi s o each peak we e ma ked on
op o each one and he alues can be compa ed wi h he heo e ical alues ob ained wi h he so wa e
Mes e-C o he molecules ske ched on he le o each plo . Fou peaks we e ob ained associa ed wi h
wo di e en ypes o in e ac ions (C-C and C-O) in each molecule (Figu e 4.9(a)). The peaks ela ed o
1,4-hyd oquinone (H2Q) we e loca ed a 150.14 ppm and 116.10 ppm showing a good ag eemen wi h
he expec ed alues a 151.42 ppm and 117.45 ppm. The wo o he peaks a 188.12 ppm and 136.98
ppm co espond wi h he expec ed alues o he 1,4-benzoquinone (Q) ( heo e ical alues a e 187.00
ppm and 135.58 ppm). On he o he hand, in he 1H spec um shown in Figu e 4.9(b) wo peaks we e
obse ed o each C-H bond in each molecule, and a collec ion o smalle peaks be ween 8.42 ppm and
8.33 ppm associa ed wi h he O-H bond. Again, he measu ed alues o he chemical shi s (6.83 ppm
o Q and 6.53 ppm o H2Q) we e in good ag eemen wi h he expec ed ones (6.90 ppm and 6.66 ppm
espec i ely).
Resul s we e simila o he expe imen s done wi h Me OH-d4. The esul s a e p esen ed in Figu e
4.10(a) o 13C and 4.10(b) o 1H. In he 13C case, he heo e ical shi s p edic ed o Q we e 135.58 ppm
o and 187.01 ppm, whe e he expe imen al ones ob ained we e 136.21 and 187.32 ppm espec i ely.
Fo H2Q, he heo e ical shi s we e 151.74 and 117.50 ppm whe e he expe imen al ones ob ained we e
115.39 and 149.81 ppm. Fo he 1H case, he heo e ical displacemen o he H-C in e ac ion in Q was
6.91 ppm compa ed o 6.77 ppm o he expe imen al shi . Fo H2Q, he chemical shi s o he H-C and
H-O in e ac ions we e 6.65 and 8.23 ppm o he heo e ical and 6.61 and 8.29 ppm o he expe imen al
80
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Figu e 4.9: NMR spec a wi h DMSO-d6as sol en o (a) 13C and (b) 1H. In bo h igu es, he expe imen al
shi s a e compa ed wi h he heo e ical p edic ions indica ed o e he schema ic molecules. As can be seen, he
expe imen al chemical shi s ag eed wi h he heo e ical ones.
cases, espec i ely. In bo h cases, he o e all ag eemen be ween he expe imen al measu emen s and he
heo e ical es ima ions was e y good wi h a maximum e o o less han 0.1 ppm.
Figu e 4.10: NMR spec a esul s o Me OH-d4sol en o (a) 13C and (b) 1H. Simila o he esul s p esen ed in
Figu e 4.9, he heo e ical and expe imen al shi s ag eed signi ican ly.
All hese esul s s ongly sugges ed ha he p ecipi a e was composed o Q and H2Q. Ne e heless,
bo h species a e key chemical in e media ies o he BZ-CHD eac ion ha play a majo ole in he
dynamics o eac ion as de ailed in Szalai e al [189]. Howe e , when Q and H2Q each ce ain le els
o concen a ion, a non-soluble chemical complex, known as quinhyd one (o Benzoquinhyd one -
Q·H2Q) can be o med by he elec onic a ac ion be ween he a oma ic ings o Q and H2Q [37].
A ep esen a ion o he molecula s uc u e o he complex quinhyd one is p esen ed in Figu e 4.11.
As was de ailed in Sec ion 1.8.3, many ins abili ies such as hose induced by eac ions o ype A
+ B →C, occu due o he di e en ial densi y be ween p oduc C and eac an s A and B. Howe e , in
his case he eac ion is no elemen a y and se e al eac an s and p oduc s a e in ol ed. The p ecipi a e
quinhyd one can be conside ed a dense p oduc capable o induce he inge ing ins abili y wi h a simila
mechanism o hose simple cases. Ne e heless, due o he complexi y o he eac ion, he sys em has
now many di e en con ol poin s ha can add ichness o he phenomenon.
81
Chap e 4. De ailed Chemical Analysis
Figu e 4.11: Chemical s uc u e ep esen a ion o he complex quinhyd one.
4.5 Chap e Discussion
In he p esen chap e , many con ol expe imen s we e p esen ed in o de o elucida e he mechanism
o he inge ing ins abili y p oduced by he BZ-CHD eac ion.
The de ailed chemical analysis was use ul o isola e e e y possible con ol poin o he sys em
es ablishing he main species in ol ed in he obse ed phenomena.
The addi ion o sodium chlo ide inhibi ed he ac ion o he ca alys leading he sys em in o a
unca alyzed s a e. This expe imen s showed ha inge ing ins abili y occu ed e en wi hou he ca alys .
This sugges ed ha he co e mechanism o he BZ-CHD eac ion was esponsible o induce he ins abili y.
The expe imen s done wi hou CHD showed a single chemical on p oduced by he oxida ion o
he e oin in o e iin. In such expe imen s, no inge ing was obse ed. This sugges ed ha he CHD
(o i s de i a i es) is a key species in he de elopmen o he ins abili y. This esul also con i med he
expe imen s done by adding NaCl. As he e oin/ e iin swi ch was no capable o induce he ins abili y,
his comple ely demons a ed ha such species was no in ol ed in he inge ing phenomenon.
The esul s ob ained om he UV-Vis spec a demons a ed he quali a i e equi alence be ween
he chemical obse ables ob ained om he con ec i e expe imen s wi h hose ob ained om he ba ch
sys em. In bo h cases, he inc emen o he exci abili y dec eased he chemical induc ion imes and
pe iod, demons a ing he ole played by chemis y in he con ec i e sys em. The ex ensi e chemical
analysis allowed o co ela e he p ecipi a e obse ed in he Hele-Shaw cell wi h he one obse ed in a
ba ch eac o .
The NMR spec oscopy was used as a he de ini i e ool o un eil he chemical s uc u e o he
p ecipi a e, he quinhyd one (Q·H2Q) complex. This species was gene a ed by he complexa ion o
wo componen s al eady p esen in he BZ-CHD eac ion mix u e, Q and H2Q. Bo h a e key eac ion
in e media ies ob ained om he oxida ion o he CHD by he B O3–and he ca alys as shown in
p e ious wo k done by Szalai e al [189]. The yellow colo a ion obse ed in he unca alyzed con ec i e
sys em was also indica i e o he p esence o 1,4-benzoquinone (Q) in he medium, which esul ed in an
addi ional con i ma ion o he p oposed mechanism.
All he in o ma ion ob ained was c ucial o de elop a sui able model capable o ep oduce he
beha io and dynamics obse ed in he con ec i e sys em. This opic will be add essed in he ollowing
chap e .
82
Chap e 5
Nume ical Resul s
Abs ac : The p e ious chap e showed ha he p ecipi a e obse ed in bo h, he
con ec i e and ba ch sys ems was quinhyd one. This compound is gene a ed by he
complexa ion o Q and H2Q. The esul s sugges ed ha he ins abili y may be p oduced
by his species h ough a classical A + B →C like mechanism. To p o e his hypo hesis,
non-linea nume ical simula ions we e done o ob ain compa able esul s. In his con ex ,
he p esen chap e will s udy he equi alence be ween he homogeneous sys em and
he a ailable kine ics models. Once known he op imum se o pa ame e s ha be e
ep esen he expe imen al esul s, he BZ-CHD kine ic models in oduced by Szalai e al
[188, 187, 189, 190] will be adap ed o a spa ially ex ended con igu a ion o simula e he
eac ion-di usion and eac ion-di usion-con ec ion sys ems. All hese nume ical models
will be used as he de ini i e ool o demons a e he mechanism o he ins abili y. The
main quan i a i e esul s p esen ed in he expe imen al sec ion we e ecalcula ed om he
simula ions and compa ed wi h he expe imen al alues o show he ag eemen be ween
hem.
5.1 Equi alence be ween Expe imen s and Reac ion Models
5.1.1 Quali a i e Compa ison
The esul s ob ained by spec oscopy we e di ec ly compa ed wi h he ex ended kine ic model
shown in Table 2.8. Howe e , due o he di e ences be ween he model and expe imen s, i was
con enien o exp ess he nume ical exci abili y as he ini ial acid concen a ions. I was ound ha he
nume ical exci abili ies ha be e i he expe imen al alues ε= 0.207, 0.826, and 1.653 M we e [H+]0
= 3 M o he lowe , [H+]0= 10 M, o he middle, and [H+]0= 15 M o he highe cases espec i ely.
These alues we e es ima ed by i ing he expe imen al alues wi h he simula ions using he GNU
so wa e COPASI.
Figu e 5.1 p esen s a compa ison be ween he nume ical and expe imen al empo al dynamics o
h ee di e en exci abili ies. As can be obse ed, he model induc ion ime and he ampli ude o he
oscilla o y egion dec eased in e sely o he exci abili y simila ly o he expe imen s. The absence o
oscilla ions in he low exci abili y case (ε= 0.207 M) was also obse ed in he nume ical simula ions.
Also, he las long-pe iod oscilla ion obse ed in he expe imen al cases be o e he beginning o he
sa u a ion egion was also ep oduced by he model.
The oscilla o y dynamics o he sys em was also well ep oduced by he model. Figu e 5.2 compa es
he simula ions esul s (Figu e 5.1(a)) wi h he expe imen s (Fig. 5.2(b)), o a high exci abili y case
[H+]0= 15 M and ε= 1.653 M espec i ely. In bo h cases, he oscilla o y egion is ma ked inside
Chap e 5. Nume ical Resul s
domain. ρmand κa e he densi y o he mix u e and he pe meabili y espec i ely (bo h dependen on
he chemical concen a ions).
Fo he sake o simplici y, he p ecipi a e was included in he model as a seconda y luid ins ead o
disc e e solid pa icles. F om he wo-phase low heo y [157, 124], he densi y o a wo-phase mix u e
can be modeled as:
ρm(Ci) = βlρl(Ci)+βsρs(5.3)
whe e βland βsa e he olume ac ions o he liquid phase and he solid phase espec i ely and βl+βs
= 1.
The densi y o he liquid phase was assumed o a y linea ly wi h espec he chemical species
concen a ions [197]:
ρl(Ci) = ρ0[1+∑
i
αiCi](5.4)
whe e,
αi=1
ρ0
∂ρ
∂Ci
(5.5)
a e he solu al expansion coe icien s [197] o each eagen (excep he [Q·H2Q]) and ρ0is he sol en
densi y.
Conside ing βs=1-βland exp essing he liquid olume as Vl=VT–Vs, whe e VTis he o al
olume and Vsis he solid olume espec i ely, he densi y o he mix u e was exp essed as a unc ion o
he olume ac ions as:
ρm(Ci) = (VT−Vs
VT
)ρl(Ci)+(1−VT−Vs
VT
)ρs(5.6)
addi ionally, Vswas also exp essed as a unc ion o he quinhyd one mola concen a ion as:
Vs=γ[Q·H2Q]RMMQ.H2QVT]
ρs
(5.7)
whe e [Q·H2Q] is he quinhyd one mola concen a ion, RMMQ.H2Qis he ela i e mola mass o he
quinhyd one and γis an ad-hoc pa ame e con enien ly se o adjus he solid ac ion o a sui able
alue o igge he inge ing ins abili y in a easonable compu a ional ime. Fo a ixed γ, he quali a i e
beha io o he expe imen al sys em was ep oduced by only changing he sys em exci abili y and
densi y jump in he same way as in he expe imen s.
Rega ding he pe meabili y κ, as explained in Shukla e al [176], he o ma ion o a p ecipi a e
a ec s he pe meabili y o he po ous ma ix whe e i is o med. Simila o he ci ed wo k, he
pe meabili y κ=κ([Q·H2Q]) was de ined as:
κ([Q·H2Q]) = κ0exp(−Rκ([Q·H2Q]
c0
)) (5.8)
whe e κ0is he pe meabili y in absence o p ecipi a e ([Q·H2Q] = 0 M, κ0=a2/12).
De ining κm=κ([Q·H2Q]) =c0,Rκcan be calcula ed as Rκ= ln(κ0/κm) simila o Shukla e al
[176]. A posi i e alue o Rκindica es ha he p ecipi a e locally educes he pe meabili y o he po ous
90

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ma ix [176]. Fo he p esen simula ions, Rκ= 0.75. This pa ame e was es ima ed ad hoc acco ding o
he expe imen al obse a ions. All he emaining pa ame e s and e e ences used o simula e he RDC
sys em ha bes i he expe imen al condi ions and esul s a e summa ized in Table 2.10 o Sec ion 2.2.
5.4 Non-Linea RDC Simula ions
5.4.1 Desc ip i e Analysis
Figu e 5.8 shows a compa a i e plo be ween h ee di e en pe spec i es: e oin concen a ion,
Q·H2Q concen a ion and Densi y (Figs. 5.8(a-c) espec i ely) o he case [H+]0= 15 M (high
exci abili y condi ion) and [Na2SO4]0= 0.200 M (la ges densi y jump) o i e di e en ime s eps. The
inge onse s a ed a e he eac ion-di usion pa e n and was na u ally p oduced by he gene a ion o
Q·H2Q. Also, he inge shape and i s dynamics shown simila i ies wi h hose obse ed expe imen ally.
As can be seen, he model showed a e y good ag eemen wi h he expe imen s (Fig. 3.1)
Figu e 5.8: Compa a i e be ween (a) Fe oin concen a ion ield, (b) quinhyd one concen a ion ield and (c)
densi y ield o a simula ed case whe e [H+]0= 15 M and [Na2SO4]0= 0.200 M which is quali a i ely equi alen
o he expe imen al coun e pa whe e ∆ρ= 0.011 g/cm3and ε= 1.653 M. This Figu e was aken om Escala e
al [61].
5.4.2 Ins abili y Va ia ion as a Func ion o ∆ρand ε
In Figu e 5.9, he densi y p o ile e olu ion is shown o ou di e en simula ed scena ios. The
simula ed cases co espond wi h he ex eme cases p esen ed in Figu e 3.2, ha is, lowes and highes
exci abili y cases, and smalles and la ges densi y jumps. Two simul aneous snapsho s a e p esen ed
o each case co esponding wi h he e oin and Q·H2Q concen a ion ields, analyzed a he same ime.
Finge ing ins abili y was p oduced a he in e ace by a A + B →C mechanism, whe e he quinhyd one is
p oduced by he accumula ion o Q and H2Q. The sys em ini ially showed he ypical eac ion-di usion
91
Chap e 5. Nume ical Resul s
pa e ns and once he inge ing ins abili y was igge ed, he chemical wa es emained mo ing a ound
he inge s. The shape o he inge s also shown simili ude wi h hose obse ed expe imen ally.
Figu e 5.9: Tempo al a ia ion o he densi y p o ile o ou di e en cases: (a) lowes exci abili y, la ges densi y
jump ([H+]0= 3 M, ∆ρ= 0.011 g/cm3), (b) highes exci abili y, la ges densi y jump ([H+]0= 15 M, ∆ρ= 0.011
g/cm3), (c) lowes exci abili y, lowes densi y jump ([H+]0= 3 M, ∆ρ= 0.002 g/cm3) and (d) highes exci abili y,
lowes densi y jump ([H+]0= 15 M, ∆ρ= 0.002 g/cm3). Fo each case, he wo ames ep esen he concen a ion
ields a he end o each simula ion o he species e oin (uppe pic) and quinhyd one (lowe pic). The densi y
p o iles we e ob ained by aking he densi y alues ac oss he line indica ed in (a). This Figu e was aken om
Escala e al [61].
Fo a low exci abili y condi ion (Figu e 5.9(a)), he o ma ion o quinhyd one was no s ong enough
as o induce inge ing ins abili y. This ac can be app ecia ed in he densi y p o ile, whe eas he ini ial
densi y jump was la ge, he sys em became mo e s able and he small amoun o Q·H2Q gene a ed ba ely
changed he densi y as o locally des abilize he sys em. This ac was no obse ed in he Figu e 5.9(c),
whe e inge ing ins abili y occu ed. Fo his case, as he ini ial densi y jump was smalle compa ed wi h
Figu e 5.9(a), he gene a ion o quinhyd one p oduced a local des abiliza ion o he sys em, igge ing
he inge ing ins abili y. In Figu es 5.9(b) and 5.9(d), as he sys em was mo e exci able, he quinhyd one
was apidly gene a ed inducing a as e inge ing ins abili y in bo h cases. Fo he mo e s able case
(Figu e 5.9(b)), he inge onse ook longe o appea compa ed wi h Figu e 5.9(d). These esul s can be
compa ed wi h hose epo ed in Figu e 3.2 and Figu es 3.3-3.4.
5.4.3 Nume ical Measu ing Obse ables
The nume ical measu ing obse ables we e calcula ed iden ically o he expe imen al case by
measu ing he space- ime plo ob ained om he simula ions. The esul s a e p esen ed in Figu es 5.10
and 5.11. Alike he expe imen s, he chemical induc ion ime ( ind−C) dec eased when he exci abili y was
inc eased. Simila o he expe imen al case, his obse able did no depend on he ini ial densi y jump
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and i was only in luenced by he ini ial chemical condi ions. The chemical pe iod (TC) also showed a
good ag eemen wi h he expe imen s. This obse able dec eased when he exci abili y was inc eased
(Fig. 5.11(a)).
Rega ding he hyd odynamic induc ion ime ( ind−H), a ela i e inc emen o his alue was obse ed
when he hyd odynamic jump was inc eased. The changes in he inge mo phology also shown
simila i ies wi h hei expe imen al case. I was possible o see an inc emen in he inge wa eleng h
o lowe exci abili y alues ( his was be e obse ed o la ge densi y g adien s). This e ec became
less signi ican o la ge exci abili ies (Figu e 5.11(b)). All hese esul s ag eed wi h he expe imen al
obse a ions p esen ed in Figu es 3.3-3.4.
Figu e 5.10: STP compa ison be ween all he simula ed cases p esen ed 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 he chemical and he hyd odynamic induc ion imes a e indica ed. All he plo s we e
ob ained by measu ing he y-line indica ed in he Fig. 5.8(a). The space and ime dimensions a e: y= 1 cm, 0= 0
and = 220 s espec i ely. Figu e aken om Escala e al [61].
93
Chap e 5. Nume ical Resul s
Figu e 5.11: Quali a i e compa ison o he nume ical obse ables o he cases p esen ed in Figu e 5.9. These
esul s we e ob ained using he same me hodology used o Figu es 3.3-3.4 (Mo e de ails a e included in Appendix
Sec ion C.1). (a) The chemical induc ion ime ( ind−C) and he chemical pe iod (TC) we e compa ed o wo
di e en exci abili ies (gi en by [H+]0). The di e ences in pe iod and induc ion ime quali a i ely ag eed wi h
he expe imen al esul s p e iously p esen ed. (b) he inge wa eleng h λHwas compa ed o di e en si ua ions
indica ed a he bo om o each plo . Simila o he expe imen s, la ge wa eleng hs we e obse ed o lowe
exci abili ies and ice e sa. This phenomenon was less signi ican in cases whe e he densi y g adien was
inc eased, due o he inc emen in he s abili y o he sys em. This ac also ag eed wi h he expe imen al sec ion.
This Figu e was aken om Escala e al [61].
5.5 Chap e Discussion
The nume ical simula ions in oduced in he p esen chap e demons a ed he e iciency o exis ing
models o he BZ-CHD eac ion. The models we e also capable o p edic he spa io- empo al dynamics
in a speci ic expe imen al con ex .
The iabili y o he ull model o ep oduce he main dynamics o he ba ch sys em was
demons a ed. This was especially app ecia ed in he chemical cha ac e is ic imes ( ind−Cand TC) wi h
a high deg ee o eliabili y. Bo h obse ables we e well ep esen ed by he model in quali a i e and
quan i a i e e ms. In addi ion, he model p edic ed he accumula ion o Q and H2Q a e he oscilla o y
egion. This was undamen al, no only o con i m he hypo hesis made o e he occu ence o he
sa u a ion egion bu also o model he quinhyd one complex o ma ion quan i a i ely.
The skele on model ep oduced he main cha ac e is ics o he ull sys em as well. Howe e , some
di e ences we e obse ed due o he la ge simpli ica ions made o e he se o equa ions. Ne e heless,
i was shown ha he use o his simpli ied model was sui able o simula e he RD and he RDC sys ems.
The modi ica ions made o bo h models (by including he Q·H2Q o ma ion), showed a ema kable
ag eemen wi h he dynamics o he ba ch sys em obse ed h ough UV-Vis spec oscopy. This was
pa icula ly no iceable in he ull model, whe e he quinhyd one o ma ion coincided wi h he sa u a ion
94
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egion o he spec um.
The de elopmen o a con ec i e model and i s implemen a ion by non-linea simula ions was
able o “na u ally” ep oduce he inge ing ins abili y. The con ec i e and oscilla ing dynamics o
he Hele-Shaw cell expe imen s we e, a leas quali a i ely, pe ec ly ep oduced by he RDC model,
showing compa able a ia ions in all he nume ical obse ables.
The con ol simula ions showed ha he ins abili y only occu ed i he chemical model was
modi ied. This demons a ed he majo ole played by he quinhyd one o ma ion in he expe imen al
obse a ions.
95

Conclusions
This pa o he hesis p esen ed a de ailed desc ip ion o he chemically induced buoyancy-d i en
inge ing ins abili y. The sys em was ully cha ac e ized by s udying he in luence o he mos ele an
pa ame e s o he sys em, he exci abili y and he densi y g adien . The use o hese wo pa ame e s was
undamen al no only o ob ain an ex ensi e desc ip ion o he sys em bu also o un eil he mechanism
behind he ins abili y o ma ion. In his sense, he e ec i eness o handling he densi y g adien and
he exci abili y independen ly was he key o p opose a sui able hypo hesis ega ding he obse ed
phenomena.
The majo in luence o chemis y in he sys em beha io was s ongly demons a ed. The nume ical
obse ables quan i ied in he con ec i e amewo k, wi h excep ion o hose associa ed pu ely wi h he
hyd odynamics, we e conse ed in bo h, he ba ch expe imen s and he nume ical models. This was
undamen al o use such models in he sys em cha ac e iza ion.
In his sense, he ins abili y obse ed was cha ac e ized o show a pa icula beha io , exhibi ing
a high deg ee o synch oniza ion wi h he chemical dynamic. Howe e , he inge o ma ion pe se
did no speci ically depend on he oscilla o y beha io as i could be ob ained jus wi h he o ma ion
o quinhyd one in absence o oscilla ions. Ne e heless, i was obse ed ha when bo h phenomena
we e p esen , a bi-di ec ional in e ac ion be ween bo h ins abili ies occu ed. This demons a ed ha in
such a speci ic condi ion chemo-hyd odynamic syne gies occu ed ha we e p oduced by simila i ies in
imescales.
I was possible o un eil he whole p ocess o inge o ma ion as an o e simpli ied mechanism,
such as he classical A + B →C, whe e he ole o C was played by he non-soluble chemical complex
quinhyd one. The de ailed cha ac e iza ion o he sys em made i possible o p opose a modi ica ion o
he exis ing eac ion mechanism ha , once coupled wi h he hyd odynamic pa , was able o nume ically
ep oduce he chemically induced densi y-d i en inge ing ins abili y. The p oposed model, unlike he
simple A + B →C case, can be con olled by a b oad a ie y o pa ame e s, p oducing a much iche
ensemble o possible beha io s.
I was sys ema ically demons a ed, ha a hyd odynamic ins abili y can be igge ed and con olled
by a chemical eac ion a he in e ace. This phenomenon can be used o unde s and why ypical
con ec i e sys ems in Na u e do no beha e as heo e ically expec ed in pu ely hyd odynamic
en i onmen s, and why he chemical in e ac ions need o be conside ed. On he o he hand, he me hods
he e exposed can be used o design mo e sophis ica ed expe imen al sys ems whe e he hyd odynamic
ins abili ies can be con olled by a chemical eac ion wi h coun less applica ions in indus y.
PART II:
VISCOUS FINGERING INSTABILITY
DRIVEN BY pH-SHIFTING
REACTIONS
Appendix A. P epa a ion o S ock Solu ions
A.1.5 Sul u ic Acid Solu ion
The sul u ic acid (H2SO4) s ock solu ion was p epa ed om concen a ed (95-98 %, 1.840 g/cm3)
H2SO4solu ion (Sigma-Ald ich, CAS: 7664-93-9, MW: 98.08 g/mol). The s ock was ob ained by adding
gen ly 356.3 mL o acid in o 250 mL o doubly dis illed wa e . As he dissolu ion p ocess is highly
exo he mic, he lask mus be cons an ly cooled. Once i was dilu ed, he olume o he solu ion was
comple ed o he ma k (1000 mL) a oom empe a u e.
A.1.6 Sodium Chlo ide Solu ion
The sodium chlo ide solu ion was p epa ed by dilu ing 0.584 g o solid sal (Sigma-Ald ich, CAS:
7647-14-5, MW: 58.44 g/mol) in o 100 mL o doubly dis illed wa e , ob aining a inal concen a ion o
0.1 M.
A.1.7 Reac i e Mix u e: Solu ions 1 and 2
The bubble- ee ecipe o he Belouzo -Zhabo insky eac ion was sepa a ed in o wo independen
and non- eac i e solu ions. Solu ion 1 was p epa ed by mixing 2.91 ml o CHD s ock, 0.24 ml o e oin
s ock and ou di e en olumes o Na2SO4s ock: 1.3 ml, 1.55 ml, 2.05 ml, and 2.3 ml espec i ely.
Solu ion 2 was p epa ed by mixing 1.42 ml o NaB O3 s ock, and 0.24 ml o e oin s ock. In bo h
solu ions, di e en olumes o H2SO4s ock we e added in equal concen a ion o a oid acid g adien s.
Speci ically: 2 ml, 4 ml, 6 ml, and 8 ml. Finally, doubly dis illed wa e was added o ob ain a inal
olume o 15 ml o each solu ion. Solu ion 1 shows a cha ac e is ic ed colo a ion as he e oin is in a
educed s a e while solu ion 2 shows a blue colo a ion due o he oxidized s a e o he e oin ( e iin).
A.1.8 P o ocol o p epa e he BZ-Aga ose Gels
The aga ose s ock solu ion was p epa ed by dissol ing 1.5 g o Aga ose TYPE I, Low EEO (Sigma,
CAS: 9012-36-6) in o 100 mL o doubly dis illed wa e ob aining a inal concen a ion o 1.5 w %. The
dissolu ion was done in boiling wa e . Once i was dissol ed, he mix u e was kep wa m a no less han
45°C (which is close o he gela ion empe a u e).
Fo p epa ing he non- eac ing aga ose gels o he sys em p esen ed in Appendix B, wo indi idual
solu ions we e p epa ed using he same s ocks indica ed in his sec ion. Gel 1 was p epa ed by adding
1.46 mL o CHD s ock, 0.12 mL o e oin s ock, 2mL o aga ose s ock and, 3.92 mL o doubly dis illed
wa e o ob ain a inal olume o 7.5 mL. Gel 2 was p epa ed by adding 0.71 mL o b oma e s ock,
0.12 mL o e oin s ock, 2 mL o aga ose s ock, and 4.67 mL o doubly dis illed wa e o ob ain a inal
olume o 7.5 mL. In bo h cases, he aga ose was added a las o a oid p ema u e gela ion. Solu ions
we e cons an ly s i ed o acili a e a homogeneous mix u e. To illing he Pe i dish, a s oppe was added
o hal o he Pe i dish (silicone ubbe o 0.5 mm o hickness is ecommended). Once he aga ose was
added in o Solu ion 1, he mix u e was pou ed in o he o he hal o he Pe i dish. Once i was geli ied,
he s oppe was emo ed and he second solu ion was added.
Once he gel was ob ained, he exci abili y was changed by adding 2 mL o dilu ed sul u ic acid
on op o he gel ( he acid can no be added di ec ly in he solu ion o mula ions as i in e e es
wi h he gela ion p ocess. The aga ose only geli ies a neu al o basic pH). The acid was dis ibu ed
homogeneously. The expe imen s a ed once hal o he Pe i dish swi ched om ed colo a ion ( educed
s a e) in o blue colo a ion (oxidized s a e).
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A.2 S ock Solu ions Used o Pa II
A.2.1 Fo maldehyde Solu ion
The o maldehyde s ock was a comme cial concen a ed Fo malin solu ion (Sigma-Ald ich, CAS:
50-00-0, MW: 30.3 g/mol).
A.2.2 Sodium Sul i e Solu ion
The sul i e s ock was p epa ed om solid NaB O3anhyd ous sal (Sigma-Ald ich, CAS: 7757-83-7,
MW: 126.04 g/mol) by dilu ing 25.21 g o sal in o 100 mL o doubly dis illed wa e , ob aining a inal
s ock concen a ion o 2 M.
A.2.3 Poly(Ac ylic Acid) [PAA] solu ions
The PAA solu ion was p epa ed om Poly(Ac ylic Acid) (Sigma-Ald ich, CAS: 9003-01-4,
A e age M ∼4000000). The s ock was ob ained by dissol ing 1 g o PAA in o 180 mL o doubly
dis illed wa e a 80°C o acili a e solubili y. A e solubiliza ion, he mix u e was cooled down o 23°C
and he inal olume was kep a 200 mL ob aining a inal concen a ion o 0.5 w %.
Fo he con ol expe imen s, a sho -chain PAA molecule was used (M sin 450000). The s ock
solu ion was ob ained by dissol ing 5 g o PAA in o a o al olume o 100 mL o doubly dis illed wa e
ob aining a inal concen a ion o 5 w %. The dissolu ion p ocedu e was he same used o he long-chain
PAA.
A.2.4 Sodium Bisul i e Solu ion
The Bisul i e s ock solu ion was p epa ed om me abisul i e sodium sal (Na2S2O5(Sigma-Ald ich,
CAS: 7681-57-4, MW: 190.11 g/mol), by dilu ing 19.011 g o sal in o 100 mL o doubly dis illed wa e ,
ob aining a inal s ock concen a ion o 1 M.
A.2.5 Sodium Hyd oxide Solu ion
The sodium hyd oxide s ock was a comme cially a ailable NaOH solu ion 5.0 M (Sigma-Ald ich,
CAS: 1310-73-2, MW: 40.00 g/mol)
A.2.6 Gluconolac one Solu ion
The gluconolac one s ock solu ion was eshly p epa ed o each expe imen om eagen g ade
D-(+)-Gluconic acid δ-lac one (Sigma-Ald ich, CAS: 90-80-2, MW: 178.14 g/mol). The s ock was
ob ained by dilu ing 0.356 g o gluconolac one in 10 ml o doubly dis illed wa e . This solu ion has been
used always esh and in less han 300 seconds a e being p epa ed o a oid hyd olysis e ec s.
A.2.7 B omo hymol Blue Indica o
The colo indica o is a 0.4 w % hyd oalcoholic solu ion o B omo hymol blue p epa ed by
dissol ing 1 g o B omo hymol blue sodium sal powde (Sigma) in o 50 ml o a 96% e hanol solu ion
dilu ing up o a inal olume o 250 mL by adding 200 mL o doubly dis illed wa e . The C.I. shows
a yellow colo o pH alues below 6 (acidic s a e), g een colo o pH be ween 6-7 (neu al s a e), and
blue colo o pH alues abo e 7 (basic s a e).
203
Appendix A. P epa a ion o S ock Solu ions
A.2.8 Displacing Solu ion Mix u e P epa a ion
As shown in Pa II, he e is no a unique ecipe o he displacing solu ion because he sul i e
concen a ion was a ied. Howe e , a base o mula ion was used o s udy he e ec s o he o maldehyde
in he ins abili y de elopmen . This base o mula ion was p epa ed by mixing 5 mL o PAA s ock, 0.195
mL o sul i e s ock, 0.300 mL o C.I. s ock, and 0.205 mL o doubly dis illed wa e . This solu ion shows
a g een colo a ion and i s pH is be ween 6.7-7.
A.3 S ock Solu ions Used o Pa III
A.3.1 Fo maldehyde Solu ion
Same ecipe as used o Pa II.
A.3.2 Sodium Sul i e Solu ion
Same ecipe as used o Pa II.
A.3.3 Poly(Ac ylic Acid) [PAA] solu ion
Same ecipe as used o p epa e PAA M ∼4000000 o Pa II.
A.3.4 Sodium Ca bona e Solu ion
The sodium ca bona e (Na2CO3) solu ion used o he con ol expe imen 5 (C5) was p epa ed om
sodium ca bona e anhyd ous powde (Sigma-Ald ich, CAS: 497-19-8, MW: 105.99 g/mol), by dilu ing
21.19 g o powde in o 100 mL o doubly dis illed wa e ob aining a inal s ock concen a ion o 2 M.
A.3.5 B omo hymol Blue Indica o
Same ecipe as used o Pa II.
A.3.6 Gluconic Acid Solu ion (Solu ion B)
Solu ion B is a concen a ed aqueous solu ion o gluconic acid 2.0 mol/kg (∼
=1.66 M), This solu ion
was ob ained by dilu ing 17.81 g o D-(+)-Gluconic acid δ-lac one (Sigma-Ald ich, CAS: 90-80-2, MW:
178.14 g/mol) in o 50 g o doubly dis illed wa e . The solu ion was le o es a comple e day o ensu e
he ull con e sion o he gluconolac one in o gluconic acid by hyd olysis.
A.3.7 Solu ion A Mix u e P epa a ion
Solu ion A was p epa ed by mixing 5 mL PAA s ock, 0.195 mL o SO32 – s ock, 0.300 mL o C.I.
s ock, 0.150 mL o Fo maldehyde s ock, and 0.055 mL o doubly dis illed wa e . This solu ion shows a
blue colo a ion and i s pH is be ween 11.5-12.
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Appendix B
Reac ion-Di usion Sys ems
Abs ac :This Appendix will in oduce he main esul s ob ained om less complex
eac ion-di usion (RD) sys ems. These esul s we e used as guidance o de elop and
unde s and he mos complex scena io p esen ed in Pa I. In his sense, he i s obse a ion
and he analysis o he dynamic o a spa ially ex ended BZ-CHD oscilla o we e done in a
1D capilla y sys em. This s udy aimed o ob ain ele an in o ma ion abou how he ypical
oscilla o y beha io o a homogeneous sys em may be a ec ed by sepa a ing pa o he
eagen s in o wo di e en non- eac i e solu ions. Wi h hose esul s, he nex s ep was o
ex end he se up o he capilla y sys em in o a 2D non-con ec i e expe imen al se up and
o s udy he in luence o a spa ially ex ended con igu a ion in he pa e n o ma ion and
hus, o be e unde s and he ole played by he chemis y in he con ec i e sys em. In
he u he sec ions, an ex ensi e desc ip ion o hose p e ious non-con ec i e sys ems is
p esen ed, showing bo h expe imen al and nume ical me hods and esul s.
B.1 Ma e ials and Me hods
B.1.1 Capilla y Sys em
1D expe imen s we e ca ied ou in he capilla y sys em schema ized in Figu e B.1. The cylind ical
capilla y eac o was buil in bo osilica e wi h an inne diame e o 0.05 mm ± 0.01 mm, 2 ± 0.1 mm ou e
diame e , and 70 ± 1 mm leng h. The small inne diame e a oids any con ec ion o 2D eac ion-di usion
pa e n, ensu ing a pu ely 1D dynamic. To ob ain he in e acial ini ial condi ion, he ollowing me hod
was used: Fi s ly, Solu ion 1 was in oduced in o he capilla y by using capilla y o ces. Once hal - illed,
Solu ion 2 was in oduced om he same opening and in he same way un il he eac o was illed. The
expe imen s we e eco ded using a CMOS came a (PixeLink PL-B776U) connec ed o a compu e wi h
a o al expe imen al ime o 6 hou s. The species o each solu ion a e indica ed in he schema ic o Figu e
B.1.
Chemical Recipe
The ecipe used is he same p esen ed in Table 2.1. As hese expe imen s we e also pa o a se o
con ol expe imen s, jus a ew expe imen s we e pe o med co e ing low, medium, and high exci abili y
cases. Mo e speci ically, hose co esponding o ε=0.207 M, ε=0.826 M, and ε=1.653 M.
As desc ibed in he p e ious sec ion, he eac o has an inne diame e o 0.05 mm which makes
i di icul o obse e he liquids inside he capilla y e en i he e a e colo ed. Fo such a eason, he
expe imen s we e done inc easing he e oin concen a ion up o ou imes he concen a ion lis ed in
Appendix B. Reac ion-Di usion Sys ems
Figu e B.1: Capilla y eac o used in 1D eac ion-di usion expe imen s. The capilla y was buil in bo osilica e.
Bo h solu ions we e in oduced in o he sys em using capilla y o ces.
Table 2.1. E en hough he inc emen in e oin could a ec he sys em empo al dynamic, i did no
in luence he quali a i e beha io o he esul s.
B.1.2 Non-Con ec i e Aga ose-Based Sys em
A pu e 2D eac ion-di usion sys em was pe o med in a Pe i dish as p esen ed in Figu e B.2. The
Pe i dish was chosen o i s simplici y and eliabili y. To ob ain a non-con ec i e sys em, he BZ-CHD
eac ion was mixed wi h an aga ose solu ion. The expe imen s we e ca ied ou in a 7 cm diame e Pe i
dish. The aga ose solu ion was p ehea ed o a oid a as geli ica ion.
The ini ial condi ion obse ed in Figu e B.2(a) was ob ained by pu ing Solu ion 1 un il hal o
he Pe i dish. Solu ion 2 was added a e gela ion o Solu ion 1. Bo h gels we e no eac i e as he
sul u ic acid was no p esen in he ecipes (aga ose does no geli y a low pH condi ions). Howe e ,
once he ini ial gel con igu a ion was ob ained, he H2SO4was added in o he sys em om he op and
dis ibu ed homogeneously h ough he en i e he gel su ace. As he acid di uses h ough he gel, he
e oin p esen in Solu ion 2 oxidizes in o e iin, ob aining a simila sys em p esen ed in Figu e B.2(b).
Analogously o he 1D expe imen s, he 2D sys em was eco ded using a CMOS came a (PixeLink
PL-B776U) o a o al expe imen al ime o 6 hou s. Simila o he 1D case, his se o expe imen s
we e used o analyze he pa e n dynamic a he in e ace and e alua e he in luence o he chemis y in
he inge ing ins abili y obse ed in he ull con ec i e sys em. All he expe imen s ealized unde his
con igu a ion we e made wi hou eplicas as hey we e used as well as a con ol sys em. In his sense,
he same condi ions o high, middle, and low exci abili y we e s udied.
Chemical Recipe
In his case, he o iginal ecipe was sligh ly modi ied o make he sys em non-con ec i e. Thus,
2 mL o aga ose 1.5 w % was added o each solu ion o make a geli ied e sion o he con ec i e
sys em. The ecipe is p esen ed in Table B.1. The sys em exci abili y was a ied in he same manne
as he capilla y expe imen s being ε=0.207 M, ε=0.826 M, and ε=1.653 M he s udied cases. The
exci abili y was a ied by adding di e en acid concen a ions o e he Pe i dish a e geli ica ion. The
eac ion s a ed once he Fe oin on he gel 2 became Fe iin.
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Figu e B.2: (a) Schema ics o he 2D eac ion-di usion sys em buil in a Pe i dish. (b) Image o an expe imen al
Pe i dish a e he e oin con e sion in o e iin. In his expe imen , bo h solu ions we e mixed wi h aga ose 1.5
w % o p oduce a pu e 2D eac ion-di usion sys em. The eac ion s a s once he acid is homogeneously pou ed
in o he Pe i dish om he uppe side.
Recipe o Gel 1 Recipe o Gel 2
Species Concen a ion (M) Species Concen aion (M)
CHD 0.291 B O3–0.142
[Fe(phen)3]2+ 0.4x10−3[Fe(phen)3]3+ 0.4x10−3
S1 olume o Aga ose 1.5 w %: 2 mL S2 olume o Aga ose 1.5 w %: 2 mL
Gel 1 olume: 7.5 mL Gel 2 olume: 7.5 mL
Table B.1: Recipe used in he non-con ec i e aga ose 2D-Reac ion Di usion Sys em.
B.2 Nume ical Models
B.2.1 Go e ning equa ions
Bo h eac ion-di usion sys ems we e modeled using he ollowing gene al se o pa ial di e en ial
equa ions:
∂Ci
∂ =DCi∇2Ci+Ri(Ci)(B.1)
whe e Riand Cia e he ne eac ion a es and he concen a ion o he in ol ed species espec i ely.
The ne eac ion a es we e ob ained om he skele on model p esen ed in Table 2.9 including he
quinhyd one o ma ion equa ion.
Analogously o he con ec i e simula ions, he modi ied skele on model was used in place o he
207

Appendix B. Reac ion-Di usion Sys ems
ull model. This was necessa y due o he la ge numbe o memo y and esou ces needed ha make i s
use p ohibi i e. This model was a enough o ep esen he quali a i e beha io o he di usi e sys ems.
All simula ions we e done using a Fini e Volume Me hod sol e implemen ed in he comme cial CFD
so wa e Ansys Fluen ® e sion 19.2 [11, 12]. All esul s we e ob ained by using he SIMPLE sol e
coupled wi h a s i -chemis y sol e . The ime s ep was au oma ically con olled by he so wa e and
bo h space and ime, we e disc e ized using he i s -o de upwind me hod. The o al compu a ional ime
was se o = 3.5 min ( eal- ime). The di e ence be ween he expe imen al and compu a ional inal
imes is due o he simpli ica ions o he modi ied skele on model used o simula e he spa ially ex ended
sys ems.
B.2.2 1D-RD Sys em Model Se up
The capilla y sys em was simula ed using a 1.5D app oach. The nume ical domain (D) consis ed
o a ec angle o wid h Land a bi a y heigh , disc e ized by using a s uc u ed mesh o 181 elemen s
in he x-di ec ion and 1 elemen in he y-di ec ion as shown in Figu e B.3. The use o a 1.5D domain
was p e e ed o acili a e he compa ison be ween expe imen s and simula ions. Ze o di usi e lux
condi ions we e se as bounda y condi ions.
Figu e B.3: 1.5D nume ical domain used o simula e he capilla y sys em. The mesh consis s o 181 elemen s
in he x-di ec ion and 1 elemen in he y-di ec ion. This ype o domain p o ides a dynamic colo p o ile o he
species concen a ion simila o he expe imen al coun e pa wi hou inc easing he compu a ional cos .
The species spa ial con igu a ion as p esen ed in Figu e B.3 is gi en by he ollowing piecewise
unc ion:
Cai(x,0) = ([Ca]0(x)(ξ (x)+1)x<L/2
0x⩾L/2
Cbi(x,0) = (0x<L/2
[Cb]0(x)(ξ (x)+1)x⩾L/2
[H2SO4]0(x,0) = k,∀x∈D
(B.2)
whe e [Ca]0a e: [CHD]0and [Fe(phen)3]2+
0.[Cb]0a e: [NaB O3]0and [Fe(phen)3]3+
0. (x,y)is a
no mally dis ibu ed andom unc ion wi h ampli ude ξ= 10−2. The [H2SO4]0was se cons an in all he
domain.
B.2.3 2D-RD Sys em Model Se up
Fo he 2D sys em, he nume ical domain (D) consis ed o a squa ed mapped mesh o a ea L2
disc e ized by 181 elemen s in bo h di ec ions, xand y. A schema ics o he domain and he spa ial
con igu a ion o he eac an species a e p esen ed in Figu e B.4. The ini ial condi ions we e se using he
same exp ession used o he RDC simula ions (Eq. 2.2.3), excep o [Na2SO4]0, which was excluded
om he RD sys em as he densi y was no aken in o accoun . All bounda ies condi ions we e se as
ze o di usi e lux.
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Figu e B.4: 2D nume ical domain used o simula e he RD model. The mesh consis s o 181 elemen s in he
ho izon al di ec ion and 181 elemen s in he e ical di ec ion.
B.3 Expe imen al Resul s
B.3.1 1D Capilla y Sys em
The dynamics o he capilla y sys em we e obse ed h ough space- ime plo s (he ea e STP).
These plo s we e ob ained by measu ing he pixel changes ac oss a e e ence line loca ed in he luid
ese oi (line indica ed as xin Fig. B.1(a)). The h ee s udied exci abili ies a e compa ed in Figu e
B.5. As can be seen, he o e all beha io o he sys em ag ees wi h he con ec i e esul s. This was
e i ied by s udying he chemical obse ables. Bo h he induc ion ime ( ind −C) and chemical pe iod
(TC) dec eased wi h he inc emen o exci abili y. In all cases, he wa eleng h also inc eased in ime. This
was pa icula ly no iceable jus be o e he oscilla ion egion ended. These beha io s we e expec ed as
his sys em was only d i en by chemis y. The gene a ion o he quinhyd one complex can be app ecia ed
as a s ong eddish ma k o he mid and high exci abili y cases.
B.3.2 Non-Con ec i e Aga ose-Based Sys em
Figu e B.6 shows he e ec o changing he exci abili y in he aga ose based non-con ec i e
sys em. Snapsho s we e aken di ec ly om he expe imen al obse a ions. As can be seen, o he
lowe exci abili y (ε= 0.207 M, Fig. B.6(c)), he sys em did no show any signi ican beha io in he
expe imen al ime. Only he di usi e displacemen o he in e ace was obse ed. This indica es ha
such a chemical condi ion was no enough o de elop spa ial s uc u es in he es ablished expe imen al
ime ( = 6h). Fo he middle exci abili y (ε= 0.826 M, Fig. B.6(b)), a eling wa es we e obse ed
mo ing h ough he in e ace, addi ionally o he ini ial on displacemen . Finally, in he highe
exci abili y case (ε= 1.653 M, Fig. B.6(a)), wa es we e obse ed o occu as e han he p e ious
case. This also ag eed wi h he con ec i e case, whe e he inc emen o he exci abili y dec eased he
induc ion ime and he pa e n wa eleng h.
The e ec o he exci abili y on he sys em beha io is be e obse ed in he space- ime plo s
ob ained om he 2D sys em. Plo s a e p esen ed in Figu e B.7 and we e ob ained by measu ing in ime
he spa ial egion indica ed wi h yin Figu e B.6(a). As can be seen, bo h he delay in he induc ion ime
209
Appendix B. Reac ion-Di usion Sys ems
Figu e B.5: Space-Time plo s ob ained om he 1D eac ion-di usion capilla y sys em o high (ε= 1.653 M),
(b) middle (ε= 0.826 M), and (c) low (ε= 0.207 M). Figu es we e cons uc ed by aking a p o ile line ac oss
he longi udinal axis o he capilla y. In all cases: [CHD]0= 0.291 M, [B O3–]0= 0.142 M, [Fe(phen)32+]0=
[Fe(phen)33+]0= 0.4x10−3M.
and he on displacemen we e isible o middle and highe exci abili es. Unlike he 1D expe imen s,
his sys em exhibi s a much la ge ac i i y showing in e ace oscilla ions o a long ime.
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Figu e B.6: 2D eac ion-di usion expe imen s o h ee di e en exci abili y condi ions (a) ε= 1.653 M, (b) ε
= 0.826 M, and (c) ε= 0.207 M. Pa e ns simila o he ones obse ed in he con ec i e case we e obse ed o
he middle and highe exci abili y cases. The emaining ini ial concen a ions we e ixed as: [CHD]0= 0.291 M,
[B O3–]0= 0.142 M, [Fe(phen)32+]0= [Fe(phen)33+]0= 0.4x10−3M.
Figu e B.7: STPs ob ained om he non-con ec i e aga ose-based sys em, The changes in wa eleng h, induc ion
ime and on displacemen a e clea ly app ecia ed. In all cases, he emaining ini ial concen a ions we e se as:
[CHD]0= 0.291 M, [B O3–]0= 0.142 M [Fe(phen)32+]0= [Fe(phen)33+]0= 0.4x10−3M.
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Appendix C. Image Analysis Techniques
Figu e C.1: Me hod used o calcula e he chemical expe imen al obse ables (TCand ind−C) ia space- ime-plo s
(STP). These esul s we e used in Figu e 3.3. (a) Speci ies he me hodology used o calcula e by indica ing he
spa ial loca ions used o gene a e he STP (ma ked as Yin he pic). In (b), he dashed lines indica e he egion
o he STP used o calcula e he plo s obse ed in (c). ind−Cwas measu ed di ec ly om he STPs, while TC
was calcula ed using he line p o iles p esen ed in (c). Each obse able was eco ded in he same egion o ime.
Pa icula ly, TCwas measu ed once oscilla ions s a ed. 0= 0 h and = 6 h.
Figu e C.2: Me hod used o calcula e he hyd odynamic expe imen al obse ables (λHand ind−H) ia
space- ime-plo s (STP). These esul s we e used in Figu e 3.4. (a) Speci ies how he STP was calcula ed. x
indica es he spa ial loca ion used o gene a e he STP. In (b), he dashed lines indica e he egion o he STP used
o calcula e he plo s obse ed in (c). ind−Hwas measu ed di ec ly om he STP, λHwas calcula ed using he line
p o iles p esen ed in column (c). Each obse able is eco ded in he same egion o ime. 0= 0 h and = 6 h.
C.2 Analysis Me hods Used o Pa II
T adi ionally iscous inge ing pa e ns ha e been s udied using mo phological desc ip ion ools
like he ac al dimension and denis y a ea. The elec ion o a di e en me hodology lies in he ype o
pa e n gi en by he expe imen s. In his wo k, due o he speci ic aspec o he expe imen al ins abili ies,
he ci cula i y was chosen as he main ool o desc ibe he e olu ion o he inge ing pa e n. Howe e ,
o check he easibili y o his me hod, esul s we e also compa ed wi h he densi y a ea ool.
C.2.1 Ci cula i y Calcula ion
The ci cula i y was used o ob ain he quan i a i e measu emen s o bo h nume ical and
expe imen al esul s. The algo i hm calcula es he ci cula i y o he shape o a bina y mask ex ac ed
om he inge ing pa e n o med by he displacing solu ion once he ins abili y is ully de eloped.
The mask ex ac ion was pe o med by using an adap i e selec ion ool (magic wand) and was hen
con e ed in o a bina y mask by applying a h eshold unc ion. As all he expe imen s we e done using
a calib a ion ba , he ela ion be ween pixels and eal spa ial uni s is known. The ci cula i y calcula ion
was la e done au oma ically by calcula ing he pe ime e and he a ea om he expe imen al/nume ical
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inge ing pa e ns. A schema ic o his p ocedu e is p esen ed in Figu e C.3. All ci cula i y measu emen s
we e done on he inal ame o each expe imen .
Figu e C.3: Schema ic o he p ocedu e used o ob ain a ci cula i y alue om he expe imen al images. The
ci cula i y alue was calcula ed om he bina y mask ob ained ex ac ed om he expe imen al/nume ical igu e.
C.2.2 Densi y A ea Calcula ion
The densi y a ea dAis de ined as he a io be ween he a ea occupied by he inge ing pa e n (A eaP)
and he adius o he longes inge RMax and is calcula ed om he ollowing exp ession [65, 142, 139]:
dA=A eaP
πR2
Max
(C.1)
As p e iously commen ed, he esul s ob ained using he ci cula i y we e compa ed wi h hose
ob ained calcula ing dA.
Figu e C.4(a) shows he dAcalcula ed o Q = 20 mL/min. These esul s mus be compa ed wi h
he esul s o Figu e 8.6. As can be no ed, his pa ame e eco e ed he same main ea u es as he
ci cula i y. Because he pa e ns we e no e y elonga ed, he changes in densi y a ea we e es ic ed o a
small ac ion o he adius and, hus, he ac ual unce ain ies became mo e impo an . This is he eason
behind he use o he ci cula i y p esen ed in he esul s o Pa II.
This demons a ed he consis ency o he esul s independen ly on he pa ame e chosen o desc ibe
hem.
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Appendix C. Image Analysis Techniques
Figu e C.4: (a) Finge densi y a ea dAcalcula ed o Q = 20 mL/min on Figu e 8.6. Bo h esul s ag ee wi h hose
ob ained using he ci cula i y. (b) dAcalcula ed a ixed dis ance d = 108 mm as a unc ion o he Fo maldehyde]0.
C.3 Analysis Me hods Used o Pa III
C.3.1 Time-Dependen Ci cula i y Calcula ion
The p ocedu e o calcula e he ime-dependen ci cula i y is p esen ed in Figu e C.5. In his
case, e e y analyzed ame o each s udied case was p ocessed by he Hema oxylin and Eosin (H&E)
colo decon olu ion algo i hm (simila esul s may be ob ained by using o he algo i hms included in
he so wa e, howe e , he de aul algo i hm was he simples choice). Once p ocessed, h ee-colo
componen s we e ob ained. A bina y mask was hen c ea ed om one o he colo componen s. Fo
Case I, he bes esul s a e ob ained bina izing he i s colo componen (Colo 1 in he schema ics o
Fig. C.5). Fo Case II, he second colo componen was chosen (Colo 2 in he schema ics o Fig. C.5).
Once he bina y images we e ob ained, he so wa e calcula ed he ci cula i y ollowing he de ini ion.
As se e al ames o a comple e expe imen al un we e p ocessed, he e olu ion o he ci cula i y can be
plo ed as a unc ion o ime.
C.3.2 A e age Displacing P o ile
The me hodology o ob ain he a e age displacing in e ace o he di ec expe imen s sha es some
simila i ies wi h he p e ious p ocedu e and i is p esen ed in Figu e C.6.
Se e al space- ime plo s (STP, indica ed as P1,P2,...,Pn) we e ob ained by aking adial slices o
a comple e expe imen al un. These STPs we e p ocessed by using he same H&E colo decon olu ion
algo i hm used o measu e he ci cula i y (Fig. C.5). Th ee colo componen s we e ob ained o each se
o STPs. Colo componen 1 was hen selec ed and bina ized ( his componen showed he bes esul s
o he p o ile cha ac e iza ion). The ob ained se o bina y STPs was hen p ocessed by edge de ec ion
and bina y skele oniza ion algo i hms ha con e ed he image da a in o a se o nume ical cu es. F om
each se o nume ical p o iles, an a e age displacemen and i s dispe sion (s anda d de ia ion) we e
calcula ed.
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Figu e C.5: Me hodology o calcula e he ime-dependen ci cula i y o Cases I and II. The ames o e e y
expe imen al case we e p ocessed in FIJI by using a colo decon olu ion algo i hm. F om such a p ocess,
h ee-colo componen s we e ob ained o each ame. Colo I and II we e used o ob ain a bina y mask o each
ype o expe imen . F om such a mask, he ci cula i y was calcula ed as indica ed in he scheme. A ci cula i y
alue was ob ained o e e y p ocessed ame. Those alues we e hen ep esen ed in a ela i e ime scale o
compa ison.
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Appendix C. Image Analysis Techniques
Figu e C.6: P ocedu e o calcula e he a e age displacing p o ile o Case I. In his case, a s ack o n-space- ime
plo s was ob ained om an expe imen by pe o ming a adial eslice ope a ion as indica ed in he igu e. E e y
ame o such s ack was hen il e ed by a colo decon olu ion algo i hm ob aining h ee s acks, one o each colo
componen . The bina y mask ob ained om colo 1 was hen p ocessed by an edge de ec ion algo i hm ob aining
a well-de ined p o ile unc ion. A skele onize algo i hm was inally used o e ie e he space and ime coo dina es
used o s a is ical calcula ions.
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Appendix D
Supplemen a y Resul s
D.1 Resul s o Pa I
D.1.1 Solu al Expansion Coe icien Calcula ion
The solu al expansion coe icien s (αi) ep esen he change in densi y p oduced by he inc emen
in he concen a ion o a chemical species [197]. In o mal e ms, hese coe icien s a e de ined as:
αi=1
ρ0
∂ρ
∂Ci
(D.1)
whe e ρ0is he densi y o he pu e sol en . These alues we e used o simula ing he RDC sys em in
Pa I as indica ed in Sec ion 5.3.
The solu al expansion coe icien s we e calcula ed expe imen ally by measu ing he densi y
a ia ion due o he changes in he concen a ions o NaB O3, Na2SO4, [Fe(phen)3]2+/3+
0, and CHD a
a cons an empe a u e. The e iin coe icien was assumed o be he same as he e oin one. The
H+coe icien was se ad hoc. The alues we e ob ained om he i ing slope o he expe imen al
measu emen s. The da a, he i ing alues, and he coe icien o de e mina ion (R2) a e shown in Figu e
D.1.
D.1.2 RDC Model Pe meabili y Va ia ion
As i was explained in he p e ious sec ion, he p ecipi a e in e ac s wi h he po ous ma ix by
educing locally he pe meabili y [176]. This implies a posi i e Rκ ac o [176]. Figu e D.2 shows he
pe meabili y d op gene a ed due o he p ecipi a ion o ma ion. The ob ained esul s we e simila o
hose p esen ed in p e ious wo ks [176].
D.1.3 RDC Con ol Simula ion
Figu e D.3 shows a RDC con ol simula ion whe e Equa ion R1 (quinhyd one o ma ion) was no
included in he skele on model. This simula ion is in ended o demons a e ha i is no possible o ob ain
a inge ing ins abili y wi hou modi ying he o iginal kine ic model.
As can be seen, he ins abili y was no obse ed and i s beha io was simila o he RD simula ions
p esen ed in Appendix B.

Appendix D. Supplemen a y Resul s
Figu e D.1: Fi ing o he solu al expansion coe icien s ob ained om expe imen al da a o (a) NaB O3, (b)
Na2SO4, (c) [Fe(phen)3]2+/3+ and (d) CHD.
Figu e D.2: Pe meabili y as a unc ion o he p ecipi a e concen a ion o [H+]0= 15 M and ∆ρ= 0.011 g/cm3.
A = 0 s, he pe meabili y inside he nume ical domain was cons an . Once he p ecipi a e s a ed o ming, he
pe meabili y go educed a i s icini y. The plo was ob ained by measu ing he pe meabili y ac oss a line aligned
o he y-di ec ion in he RDC simula ions. This Figu e was aken om Escala e al [61].
224
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Figu e D.3: 2D simula ions o he skele on model plus he equa ions desc ibing he con ec ion in he sys em o
[H+]0= 15 M, ∆ρ= 0.002 g/cm3. As he model does no include he gene a ion o quinhyd one, i is unable o
ep oduce he inge ing ins abili y and a plana in e ace emains in he sys em. This Figu e was aken om Escala
e al [61]
225
Appendix D. Supplemen a y Resul s
D.2 Resul s o Pa II
D.2.1 E ec o he Colo Indica o
E en hough he colo indica o acili a es he expe imen al obse a ion and he image p ocessing,
i was necessa y o e i y i i had any in luence on he p esen ed esul s. In ha sense, he e ec on he
heology o he sys em due o he colo indica o was measu ed.
In Figu e D.4, a compa ison be ween wo di e en samples used in he main expe imen s is
p esen ed. On he one hand, he sample indica ed as “Wi h C.I.” co esponds o a solu ion con aining
0.438 w % o PAA, 0.068 M o SO32 – , and 0.021 w % o C.I. On he o he hand, he cu e indica ed as
“Wi hou C.I.”, co esponds o a solu ion in which he colo indica o was eplaced by doubly dis illed
wa e . The emaining eagen s we e kep equal as he colo ed case.
Bo h cu es we e compa ed in a iscosi y-shea a e plo measu ed by he heology equipmen
desc ibed in Sec ion 6.1.2. As can be seen, no signi ican change was obse ed in he iscosi y o he
solu ion due o he addi ion o he colo indica o . This demons a es ha in he used concen a ions his
compound did no a ec he heology o he sys em.
Figu e D.4: S udy o he e ec p oduced by he colo indica o in he o e all iscosi y o he displacing solu ion.
The black do ed cu e co esponds o he iscosi y o he displacing solu ion used in he colo ed expe imen , while
he ed squa ed cu e co esponds o he iscosi y o he solu ion used in he Schlie en expe imen s (Fig. 8.8). This
demons a es ha he heology o he sys em was no a ec ed by he addi ion o he colo indica o . This Figu e
was adap ed om Escala e al [58].
D.2.2 Elas ici y E ec s and Shea Ra e Es ima ion
Non-New onian luids, like he polyme ic solu ions used in his wo k, show many in e es ing
cha ac e is ics. One o hem is he dependence o he iscosi y wi h he shea a e. In pa icula , he PAA
shows a shea - hinning New onian beha io . This means ha he iscosi y o a PAA aqueous solu ion
dec eases by inc easing he shea a e.
Ano he impo an cha ac e is ic is he elas ici y. The elas ic p ope y o a polyme ic solu ion is
also ela ed wi h o o e lap concen a ion (c∗), which was in oduced in Sec ions 1.9 and 7.6. The s udy
o he elas ici y o he PAA solu ions is undamen al o disca d any undesi ed a i ac ela ed o i and
226
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o ensu e ha all esul s obse ed we e d i en jus by he changes in iscosi y p oduced by he s udied
chemical eac ions [139, 140].
In simila wo ks, he elas ici y o simila PAA solu ions was s udied in a simpli ied manne by
measu ing he i s no mal s ess di e ence (N1) [139]. No mal S esses a e caused by shea o ces
and a e ypically obse ed in polyme solu ions [70, 211, 215]. Such s esses may appea no only in
heological measu emen s (especially when liquids a e con ined in a cone-pla e geome y as he one used
in Chap e 7) bu also inside a Hele-Shaw cell. The magni ude o he no mal s ess depends no only on
he ype o luid bu also on he shea a e o which ha luid is exposed. Fo his eason, i is undamen al
o s udy he elas ici y o he PAA solu ions in he ange o shea a es used. Fo all esul s p esen ed in
Chap e 7, he shea a e was ixed o a alue o 500 s−1. Howe e , inside he Hele-Shaw cell, he shea
a e is no cons an and had o be measu ed.
Based on he p e ious wo k o Naga su e al [140, 142, 139], he shea a e inside a adial Hele-Shaw
cell can be es ima ed as:
˙
γ =Q
π ia2(D.2)
whe e i= 6.18 cm is he adius o he ini ial condi ion ob ained om he expe imen s, and a= 0.25 mm
is he gap be ween bo h pla es.
By conside ing he ex eme cases (Q= 2.5 - 10 mL/min), he es ima ed ange o shea a e a he
icini y o he inge ip was:
3.44 ≤˙
γ [s−1]≤27.48 (D.3)
The measu emen o N1was done by using he same heological equipmen p esen ed in Sec ion
6.1.2. The mos concen a ed solu ions (and expec ed o be he mos elas ic ones) shown in Figu es
7.3(b) and 7.8(a), we e compa ed wi h he con ol solu ion o PAA and NaOH. The esul s a e p esen ed
in Figu e D.5.
As can be seen in Figu e D.5(a)), he con ol solu ion (g een diamonds cu e) exhibi s a s ong
inc ease in no mal s ess o ˙
γ >100 s−1. This was expec ed since a 4x106gmol−10.47 w % PAA
solu ion geli ies o a b oad ange o NaOH concen a ions due o he ex ension o he polyme chain
[142]. As was discussed in Sec ion 7.6, he PAA dissocia ion inc eased he adius o gy a ion Rgand in
consequence, an o e lapping scena io was eached.
Howe e , he blue do ed cu e shows ha he e was no measu able no mal s ess ound in he
ange o shea a es used o bo h he Hele-Shaw expe imen s and s i ed sys em (500 s−1). In his case,
e en wi h he same polyme concen a ion, he p esence o sodium ions p o ided by he SO32– solu ion
quenched he elec os a ic epulsion impeding he elonga ion o he PAA molecules. This quenching
e ec p oduced a dec emen in he elas ic p ope y and he e o e no o e lap occu s a he analyzed
concen a ion.
A simila si ua ion can be obse ed in igu e D.5(b), whe e he no mal s ess o he con ol solu ion
(g een diamonds cu e) shown a s ong inc emen o shea a es a ound 100 s−1, while he blue do ed
cu e does no exhibi any app eciable no mal s ess in he ange o s udied shea a es. This was
p oduced due o he same quenching e ec obse ed in igu e D.5(a)).
As hese esul s we e ob ained o he mos iscous (and also expec ed he mos elas ic) solu ions,
his conclusion can be ex ended o he es o he alues measu ed in Figu es 7.3-7.5 and 7.8, included
all he Hele-Shaw expe imen s.
227
Appendix E. Copy igh Pe missions
234

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