UNIVERSIDAD SIMÓN BOLÍVAR DECANATO DE ESTUDIOS … · diciembre 2015 universidad simÓn bolÍvar...

DECEMBER 2015

UNIVERSIDAD SIMÓN BOLÍVAR DECANATO DE ESTUDIOS DE POSTGRADO COORDINACIÓN DE POSTGRADO EN INGENIERÍA DE MATERIALES DOCTORADO EN INGENIERÍA DE MATERIALES

UNIVERSITÉ DE PAU ET DES PAYS DE L’ADOUR UFR SCIENCE ET TECHNIQUES ÉCOLE DOCTORAL DE SCIENCES EXACTES ET LEURS APPLICATIONS SPÉCIALITÉ : PHYSICO-CHIMIE DES POLYMÈRES

DOCTORAL THESIS

EVALUATION OF AN OIL-BASED SELF-DEGRADABLE GEL FOR OIL

PRODUCTION APPLICATIONS By : Oscar Eduardo Vernáez Hernández Under an international co-tutelage of directors: Prof. Alejandro Müller and Prof. Bruno Grassl Co-director: Sylvie Dagreou

Diciembre 2015

UNIVERSIDAD SIMÓN BOLÍVAR

DECANATO DE ESTUDIOS DE POSTGRADO COORDINACIÓN DE POSTGRADO EN INGENIERÍA DE MATERIALES DOCTORADO EN INGENIERÍA DE MATERIALES

EVALUACIÓN DE UN GEL AUTODEGRADABLE

BASE ACEITE PARA APLICACIONES PETROLERAS Tesis Doctoral presentado a la Universidad Simón Bolívar por:

Oscar Eduardo Vernáez Hernández

Como requisito parcial para optar al grado académico de:

Doctor en Ingeniería de Materiales

Con la asesoría del Prof.

Alejandro Müller

En cotutela con el Prof.

Bruno Grassl y Sylvie Dagreau (Université de Pau et de Pays de l’Adour)

Décembre 2015

UNIVERSITE DE PAU ET DES PAYS DE L’ADOUR UFR SCIENCE ET TECHNIQUES École doctoral de Sciences Exactes et leurs Applications Equipe de Physique et Chimie des Polymères

THÈSE

Pour obtenir le grade de : Docteur de L’Université de PAU ET Des PAYS DE L’ADOUR Discipline : Chimie-Physique Option : Physico-chimie des polymères EVALUATION D’UN GEL AUTODÉGRADABLE À BASE

D’HUILE POUR APPLICATIONS PÉTROLIÈRES par : Oscar Eduardo Vernáez Hernández

Soutenue le 02 de Décembre 2015 à l’Université de Pau et des Pays de l’Adour Devant la commission d’examen Dominique Hourdet, professeur, Université Pierre et Marie Curie (rapporteur) Antxon Santamaría, professeur, Université du Pays Basque (rapporteur) Lionel Choplin, professeur, ENSIC Nancy (Jury) Guillaume Dupuis, docteur, Poweltec (Jury) Bruno Grassl, professeur, Université de Pau et des Pays de l’Adour (Directeur) Alejandro Müller, professeur, Université Simon Bolivar (Directeur) Sylvie Dagreou, maître de conférences, Université de Pau et des Pays de l’Adour (Co-encadrante)

Diciembre 2015

UNIVERSIDAD SIMÓN BOLÍVAR

DECANATO DE ESTUDIOS DE POSTGRADO COORDINACIÓN DE POSTGRADO EN INGENIERÍA DE MATERIALES DOCTORADO EN INGENIERÍA DE MATERIALES

EVALUACIÓN DE UN GEL AUTODEGRADABLE BASE ACEITE PARA

APLICACIONES PETROLERAS

Por: Vernáez Hernández Oscar Eduardo Carnet No: 0786232 Esta Tesis Doctoral ha sido aprobada en nombre de la Universidad Simón Bolívar por el siguiente jurado examinador: ___________________________ Presidente

__________________ Miembro Externo _____________________ Miembro principal-Tutor ____________________ Miembro Principal ____________________ Miembro Principal-Tutor

Diciembre 2015

Diciembre 2015

DEDICATION

A mi amada familia, mi esposa Bárbara y mis hijos Francisco y Paula. También a mi madre Trina. A ellos les dediqué el esfuerzo y por ellos valió la pena.

Diciembre 2015

ACKNOWLEDMENTS

I would like to thank to PDVSA Intevep, for funding most of the project through the Department of Well Productivity and for the nine month leave granted. Also FONACIT (Science and Technology Ministry of the Bolivarian Republic of Venezuela) and Ministry of Foreign Affairs (France) for funding received through project: PCP 2011001409 (Postgraduate Cooperation Project) regarding the stay in the Université de Pau et des Pays de l’Adour. I would like to express my gratitude to my mentor Alejandro Müller for his patience, dedication and advice, not only during the doctoral thesis, but also for his great influence in my professional career. Mes remerciements vont également à Bruno Grassl et Sylvie Dagreou, pour l'hospitalité dont ils ont fait preuve envers moi lors des deux séjours que j'ai effectués dans leur groupe. Je remercie tous ceux dans le groupe de travail de EPCP a l’IPREM, qui avec leur hospitalité et leur assistance ont fait mes séjours en France très agréable.

Merci à Prof. Hourdet et Prof. Santamaría qui ont accepté d'être les rapporteurs de cette thèse, et je les en remercie, de même que pour leur participation au Jury.

Egalement mes remerciements à Prof. Choplin et Dr. Dupuis pour avoir accepté de faire partie du Jury. A mis colegas y compañeros de PDVSA Intevep, los que están y los que alguna vez estuvieron. A mis padres por el apoyo incondicional que me han brindado durante toda mi vida y la suya. Y finalmente, a mi esposa e hijos, quienes estuvieron conmigo en todo momento, me apoyaron y mucho sacrificaron para que yo pudiese completar esta meta. A ellos mi agradecimiento eterno.

Diciembre 2015

i December 2015

EVALUATION OF AN OIL-BASED SELF-DEGRADABLE GEL

FOR OIL PRODUCTION APPLICATIONS

Summary

In oil well treatments, such as matrix stimulations or water shut-off, it is often necessary to temporary isolate or protect productive zones with chemical diverting agents. In this work, a solution of peroxide crosslinked styrene-butadiene rubber (SBR) has been transformed to a self-degradable gel system by adding hydroperoxide as a degradation agent to the formulation. This formulation has been characterized by dynamic rheometry. The results were employed to calculate the relaxation time spectra from dynamic moduli measurements with a new method developed for the calculation in this work. The new method is presented along with a full review of methods for calculation of the relaxation time spectrum from oscillatory rheometry. In-situ and ex-situ experiments were performed to evaluate the evolution of crosslinking and degradation reactions, including the liquid-solid transition. Structural changes in the polymer network were visible within the relaxation time spectra. The degradation kinetics of styrene butadiene rubber (SBR) in solution was studied in anaerobic conditions and the characterization was performed by multiangle light scattering coupled to size exclusion chromatography (SEC-MALS). Using population balance equations, it was possible to calculate the kinetic constants for thermal and thermooxidative degradation. Analysis of the degradation results led to the conclusion that random scission of polymer chains produced by macroradicals formed by hydrogen abstraction constituted the predominant SBR degradation mechanism. Different formulations of these oil-based self-degradable gels have been evaluated as possible diverting agents during oil wells operations. The time dependent rheological behavior of the gels was measured to verify polymer crosslinking, maximum gel strength and gel degradation, all of which can be adjusted by varying formulation depending on operation needs. Single core tests were performed to evaluate pressure resistance as a function of gel strength and gel degradation. Parallel cores tests were also carried out to validate diversion efficiency and mobility restoration.

ii Diciembre 2015

EVALUACIÓN DE UN GEL AUTODEGRADABLE BASE ACEITE

PARA APLICACIONES PETROLERAS

Resumen

Para el desarrollo de un nuevo agente de divergencia química para la industria petrolera, se estudió un sistema polimérico gelificante a base de caucho estireno butadieno (SBR), solubilizado en aceite que puede ser entrecruzado con peróxidos orgánicos en presencia de temperatura y que además puede auto-degradarse por efecto de la temperatura en tiempos controlables mediante la adición de un hidroperóxido orgánico. La caracterización del sistema se realizó mediante los cambios reológicos de las muestras durante la transición líquido-sólido tanto en el proceso de entrecruzamiento como en el proceso de degradación. Se presenta una revisión teórica con los diferentes métodos para el cálculo del espectro de relajación a partir de datos experimentales. Se desarrolló un método propio para el cálculo del espectro de relajación característico que pudiese obtenerse a partir de ensayos oscilatorios con un corto rango de frecuencia experimental y que permite caracterizar los cambios reológicos del sistema, de donde se obtuvieron espectros continuos con modos de relajación relacionados con los cambios estructurales del sistema durante el entrecruzamiento y la degradación. Se presenta un estudio de la degradación del polímero en solución en presencia de hidroperóxido en condiciones anaeróbicas, que fue caracterizado mediante el uso de ecuaciones de balance poblacional. Por último, se realizaron ensayos de simulaciones físicas en núcleos de medios porosos para evaluar la formación y degradación del gel en matrices porosas, su resistencia a la presión y la restauración de la permeabilidad por degradación del gel, de donde se concluyó que el gel desarrollado es apto para ser aplicado en pozos petroleros como agente de divergencia química.

iii Décembre 2015

EVALUATION D’UN GEL AUTODÉGRADABLE À BASE

D’HUILE POUR APPLICATIONS PÉTROLIÈRES

Résumé

Dans les opérations de traitement de puits de pétrole, comme la stimulation au sein de roches, la prévention des venues d’eau, il est souvent nécessaire d'isoler ou protéger temporairement des zones de production avec agents chimiques de dérivation. Dans cette thèse, une solution de caoutchouc de styrène-butadiène (SBR) réticulé avec du peroxyde organique, a été transformé en un gel autodégradable par l’ajout d’hydropéroxyde comme agent de dégradation. La formulation a été caractérisée en rhéologie dynamique. Les résultats ont été utilisés pour calculer les spectres de temps de relaxation à partir des modules dynamiques de cisaillement, par une nouvelle méthode développée dans ces travaux. Cette nouvelle méthodologie est présentée avec une revue complète de méthodes pour le calcul des spectres de relaxation à partir de données de rhéométrie de cisaillement. Des expériences in-situ et ex-situ ont été réalisées afin d’évaluer l’évolution de les réactions de réticulation et dégradation, y compris la transition liquide-solide. Les changements de la structure du réseau polymérique sont mises en évidence dans les spectres de temps de relaxation. La cinétique de la dégradation de la solution de SBR a été étudiée dans des conditions anaérobies et caractérisée par diffusion de la lumière multi-angles couplée à la chromatographie d’exclusion stérique (SEC-MALS). En utilisant des équations de bilan de populations, nous avons pu calculer les constantes cinétiques de dégradation thermique et thermo-oxydative. L’analyse des résultats de dégradation a conduit à la conclusion que la scission aléatoire des chaînes de polymère, due aux macro-radicaux formés par abstraction d’hydrogène, est le mécanisme prédominant de la dégradation du SBR. Différentes formulations de ces gels auto-dégradables à base d’huile ont été évaluées en tant que potentiels agents de dérivation pour des opérations de traitement de puits. Le comportement rhéologique de ces gels a été mesuré en fonction du temps, pour vérifier la réticulation du polymère, la force de gel, et la dégradation ; ces variables peuvent être ajustées aux nécessités de l’opération de traitement en jouant sur la formulation. Des tests d’écoulement dans carotte simples ont été réalisés pour évaluer la résistance à la pression en fonction de la force du gel et de sa dégradation. Des tests d’écoulement dans carottes parallèles ont également été menés, afin d’évaluer l’efficacité de la dérivation et la restauration de mobilité.

iv December 2015

TABLE OF CONTENTS

LIST OF FIGURES ......................................................................................................................... xi

INTRODUCTION ............................................................................................................................ 1

CHAPTER I

THE RELAXATION TIME SPECTRUM FROM RELAXATION DATA: A THEORETICAL REVIEW .............................................................................................................. 6

ABSTRACT ...................................................................................................................................... 7

RESUMEN ........................................................................................................................................ 8

RÉSUMÉ ........................................................................................................................................... 9

Introduction ............................................................................................................................... 10

Maxwell models for viscoelastic materials ..................................................................... 13

Relaxation time spectrum ..................................................................................................... 16

Ill-posed problems ................................................................................................................... 18

Approximation methods ........................................................................................................ 19 First approximation method .................................................................................................................. 19 Higher derivative methods ..................................................................................................................... 20 Power law approximation ....................................................................................................................... 22 Matrix decomposition Method .............................................................................................................. 23 Laplace method ............................................................................................................................................ 24 Padé-Laplace Method ................................................................................................................................ 25 Emri-Tschoegl Algorithm ........................................................................................................................ 28 Winter-Chambom spectrum ................................................................................................................... 29

Minimization methods: .......................................................................................................... 31

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Cost Functions .............................................................................................................................................. 31 Parsimonious discrete spectrum IRIS® ............................................................................................. 33

BSW Spectrum ................................................................................................................................................... 34 Regularization methods ........................................................................................................................... 36 Singular value decomposition ............................................................................................................... 38 Maximum entropy method ..................................................................................................................... 40 CONTIN algorithm ...................................................................................................................................... 41 Non-linear regularization methods (NLREG) ................................................................................. 42 KKT minimization ...................................................................................................................................... 44 Bayesian analysis ........................................................................................................................................ 44 Choosing regularization parameter .................................................................................................... 45 Simulated annealing .................................................................................................................................. 47 Linearization Method ................................................................................................................................ 49 Smoothness factor ...................................................................................................................................... 50 Cubic Hermite Spline ................................................................................................................................. 51 Wavelet Transform (WT) method ....................................................................................................... 52 Fixed point iteration method ................................................................................................................. 54 Power series iterative method .............................................................................................................. 56 BLM method .................................................................................................................................................. 58

Relaxation time range ............................................................................................................. 59

Final comments ......................................................................................................................... 61

Acknowledgments .................................................................................................................... 62

References .................................................................................................................................. 62

List of Symbols ........................................................................................................................... 70

vi December 2015

CHAPTER II

Relaxation Time Spectra from Short Frequency Range Small Angle Dynamic Rheometry ................................................................................................................ 74

ABSTRACT ................................................................................................................................... 75

RESUMEN ..................................................................................................................................... 76

RÉSUMÉ ........................................................................................................................................ 77

Introduction ............................................................................................................................... 78

Proposed Method ..................................................................................................................... 85 Linearization ................................................................................................................................................. 85 Regularization .............................................................................................................................................. 87 Relaxation time range ............................................................................................................................... 89 Minimization ................................................................................................................................................. 91 Regularization parameter ....................................................................................................................... 93 Spectrum normalization .......................................................................................................................... 94

Application of the method for simulated spectra ......................................................... 95 Smoothing and interpolation ................................................................................................................. 99

Pseudo-code ............................................................................................................................. 101

Experimental frequency range .......................................................................................... 102

Relaxation time range ........................................................................................................... 105

Relaxation spectra from experimental data ................................................................. 107

Conclusions ............................................................................................................................... 111

References ................................................................................................................................ 112

List of Symbols ......................................................................................................................... 115

vii December 2015

CHAPTER III

Dynamic Rheology and Relaxation Time Spectra of Oil-Based Self-degradable Gels ....................................................................................................................... 117

ABSTRACT ................................................................................................................................. 118

RESUMEN ................................................................................................................................... 119

RÉSUMÉ ...................................................................................................................................... 120

Introduction ............................................................................................................................. 121

EXPERIMENTAL ....................................................................................................................... 124 Materials ...................................................................................................................................................... 124 Methods ........................................................................................................................................................ 125

RESULTS AND DISCUSSIONS................................................................................................ 126 Effect of crosslinker ................................................................................................................................ 134 Effect of polymer concentration ........................................................................................................ 136 Effect of temperature ............................................................................................................................. 138 Effect of breaker concentration ......................................................................................................... 140 Effect of transfer agent .......................................................................................................................... 142

Conclusions ............................................................................................................................... 146

Acknowledgements ............................................................................................................... 146

References ................................................................................................................................ 146

viii December 2015

CHAPTER IV

Degradation of StyreneButadiene Rubber (SBR) in Anaerobic Conditions ................................................................................................................................. 150

ABSTRACT ................................................................................................................................. 151

RESUMEN ................................................................................................................................... 152

RÉSUMÉ ...................................................................................................................................... 153

Introduction ............................................................................................................................. 154

Model description .................................................................................................................. 156

Experimental ............................................................................................................................ 161 Materials ...................................................................................................................................................... 161 Degradation reaction .............................................................................................................................. 162 Caracterization procedures ................................................................................................................. 162

Experimental results ............................................................................................................. 163 Reproducibility and repeatability ..................................................................................................... 163 Degradation results ................................................................................................................................. 164

Discussion of the results ...................................................................................................... 169 PBE for thermal degradation .............................................................................................................. 172

(4.21) .................................................................................................................................................................. 173 PBE for degradation with hydroperoxide ..................................................................................... 174

Conclusions ............................................................................................................................... 181

Acknowledgements ............................................................................................................... 181

References ................................................................................................................................ 182

ix December 2015

CHAPTER V

Oil-based self-degradable gels as diverting agents for oil well operations ................................................................................................................................. 184

ABSTRACT ................................................................................................................................. 185

RESUMEN ................................................................................................................................... 186

RÉSUMÉ ...................................................................................................................................... 187

Introduction ............................................................................................................................. 188

Experimental ............................................................................................................................ 190 Materials ...................................................................................................................................................... 190 Equipments ................................................................................................................................................. 191 Methods ........................................................................................................................................................ 192

Single core experiments .............................................................................................................................. 192

Parallel cores experiments ........................................................................................................................ 193

Remedial application ................................................................................................................................... 193

Compatibility test .......................................................................................................................................... 194

Results and discussion ......................................................................................................... 195 Chemical crosslink and degradation reactions ........................................................................... 195 Single core experiments ........................................................................................................................ 198 Effect of initial core saturation ........................................................................................................... 200 Two parallel cores test and stimulation simulation ................................................................. 201 Field Application design ........................................................................................................................ 203 Scaling factors and productivity recovery .................................................................................... 204 Remedial test ............................................................................................................................................. 209 Compatibility tests ................................................................................................................................... 210

x December 2015

Conclusions ............................................................................................................................... 211

Acknowledgment .................................................................................................................... 212

References ................................................................................................................................ 212

Overall Conclusion ................................................................................................................. 214

xi December 2015

LIST OF FIGURES

Chapter 2

Figure 2.1a). L curve method for progressive iteration of α b) Modified L-curve method proposed in this work for non-progressive iteration of α. ........................ 94 Figure 2.2. Storage and loss moduli generated from the simulated spectrum using equations (2.3) and (2.4).. .......................................................................................................... 96 Figure 2.3. Simulated spectrum from Honerkamp and Weese (1989) and the calculated from the generated data using the method proposed in this work. .............. 97 Figure 2.4. Modified L curve to determine the optimum regularization parameter ...................................................................................................................................................... 98 Figure 2.5. Standard deviation as a function of the number of Maxwell elements for the two regularization operator ............................................................................... 99 Figure 2.6. Effect of the number m of data points in the spectrum calculation ............ 100 Figure 2.7. Truncated data generated from the simulated spectrum using Equations (2.3) and (2.4). Filled symbols are G’ and open symbols are G’’. ................... 103 Figure 2.8. Relaxation time spectrum for moduli data from the simulated spectrum, and from similar simulated spectra but truncated at three different intervals ........................................................................................................................................................ 105 Figure 2.9.Relaxation time spectra calculated for different values of B normalized with 10 Maxwell elements per decade. .................................................................. 106 Figure 2.10. Small angle dynamic rheometry for a solution of 20mM of CTAT and different concentration of a zwitterionic copolymer (PAM-Z). ................................... 110

xii December 2015

Figure 2.11. Relaxation time spectra calculated from data of a 20mM CTAT solution with different concentration of a zwitterionic copolymer ................................... 111

Chapter 3 Figure 3. 1. Scheme indicating the competitive reactions of the final self-degradable gel behavior. ....................................................................................................................... 122 Figure 3. 2. SBR radical reactions with peroxide ....................................................................... 127 Figure 3. 3. In-situ crosslinking reaction for a 1.50 wt. % SBR solution with 0.20 wt. % of dicumylperoxide at 110 °C....................................................................................... 129 Figure 3. 4. Dynamic rheological moduli as a function of time for an ex-situ reaction at 100 rad/s for a solution containing 2.50 wt. % SBR, 2.75 wt. % of tert-butyl peroxide and 0.60 wt. % of cumene hydroperoxide ............................................ 130 Figure 3. 5. Frequency sweeps for ex-situ experiments at different reaction times.. ............................................................................................................................................................ 132 Figure 3. 6. Relaxation time spectra for a reversible gel calculated at different reaction times in ex-situ experiments............................................................................................. 133 Figure 3. 7. In-situ reaction evaluated at 7.28 rad/s for a formulation of 1.50 wt. % polymer, and 110 °C ................................................................................................................... 135 Figure 3. 8. Left: Complex modulus for a formulation with 2.5% SBR, 149 °C and 0.6 wt. % of breaker. Right: Relaxation time spectra for the tree formulations at 480 minutes. .............................................................................................................. 136 Figure 3. 9. In-situ evaluation of a formulation of 0.20 wt.% of crosslinker, at 110 °C and different polymer concentrations ............................................................................. 137

xiii December 2015

Figure 3. 10. Left: Ex-situ essays for two formulations with different polymer concentration and 2.75 wt. % of crosslinker and 0.6 wt. % of breaker. Right: Relaxation time spectra for both formulations at different reaction times .................... 138 Figure 3. 11. In-situ evaluation for different temperatures and the same formulation with 1.50 wt. % of SBR and 0.20 wt. % of dicumyl peroxide. ..................... 139 Figure 3. 12. Left: Ex-situ experiments at three different temperatures for a formulation with 2.50 wt. % SBR, 2.75 wt. % crosslinker and 0.60 wt. % hydroperoxide. Right: corresponding relaxation time spectra at their maximum gel strength ........................................................................................................................... 140 Figure 3. 13. In-situ reversible crosslinking reaction for a formulation with 1.50 wt. % of SBR at 110 ºC and different crosslinker (DCP) and breaker (HPC) concentration ............................................................................................................................... 141 Figure 3. 14. In-situ evaluation of four formulation with 0.20 wt.% of Dicumyl peroxide, 1.50 wt.% of SBR at 110 °C and different Copesol® concentration in solvent ........................................................................................................................................................... 143 Figure 3. 15. Relaxation time spectra for formulation 110 °C, 0.20 wt. % of Dicumyl peroxide, and 1.50 wt. % of SBR. .................................................................................... 144 Figure 3. 16. Ex-situ reaction for gel formulation of 2.65 wt. % of polymer, 3.00 wt. % of crosslinker, 0.10 wt. % of breaker at 149ºC and different Copesol® concentration ....................................................................................................................... 145 Figure 3. 17. Transfer reaction proposed for Copesol® and primary radicals ............. 145

xiv December 2015

Chapter 4 Figure 4. 1. a) β Scission of PB from allylic radical(Jiang, Levchik, Levchik, Wilkie 1999) b) Reaction scheme proposed for thermal decomposition of PB(Jiang, Levchik, Levchik, Wilkie 1999) c) Mechanism for free radical degradation of Polystyrene(Gordon Cameron, Meyer, McWalter 1978) ......................... 155 Figure 4. 2. Molar mass distribution at two different times .................................................. 157 Figure 4. 3. Reproducibility and repeatability tests for SBR samples in solution at initial time (before degradation). The middle solid line represents the average Mw and dash lines represent one standard deviation.................................... 164 Figure 4. 4. Distribution of molecular weights for a degradation reaction of a SBR sample in solution with 0.6 % of hydroperoxide at 120 °C. The inset shows how the Mw changes with reaction time ......................................................................... 165 Figure 4. 5. Thermal degradation experiments of SBR in solution for three different temperatures, without the use of hydroperoxide. The change in Mw was normalized with respect to the initial Mw of the sample .............................................. 166 Figure 4. 6. Degradation of SBR in solutions with 0.50 % of cumene hydroperoxide at three different temperatures. ........................................................................ 167 Figure 4. 7. SBR degradation at different hydroperoxide concentration. a): 100 ºC. b): 120ºC ...................................................................................................................................... 168 Figure 4. 8. Degradation of SBR in solution with different concentrations of an aromatic solvent (Alkylbenzene or Copesol). The experiments were performed at 100 ºC and with a concentration of hydroperoxide of 0.5% .................... 169

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Figure 4. 9. Calculation of the thermal degradation kinetic constants by PBE for three different temperatures. ...................................................................................................... 173 Figure 4. 10. Ka calculation for SBR degradation at 100 ºC and different concentrations of hydroperoxide from equation (4.38), the dashed line is the fitting until 15000 seconds. The solid line represents the fit until 23000 seconds. ........................................................................................................................................................ 177 Figure 4. 11. Ka and Kb calculation for SBR degradation at 100ºC and different concentrations of hydroperoxide from Equation 36 ................................................................ 178 Figure 4. 12. PBE approximation for SBR degradation at 120ºC. ....................................... 179 Figure 4. 13. Curve fitting for the SBR degradation with 0.50% of hydroperoxide at 100 ºC and different concentrations of alkylbenzene in the solvent. .......................................................................................................................................................... 180

Chapter 5 Figure 5. 1. Core plug permeameter scheme for parallel and single coreflood tests ................................................................................................................................................................ 191 Figure 5. 2. Viscosity behavior for a formulation with 2.50 wt. % of polymer at different temperatures ..................................................................................................................... 196 Figure 5. 3. Left: Rheological behavior of a gel formulation at 10 rad/s. Right: Frequency sweep obtained by small angle oscillatory rheometry ..................................... 197 Figure 5. 4. Normalized pressure drops for different maximum gel strengths expressed as complex modulus at a frequency of 10 rad/s................................................... 199

xvi December 2015

Figure 5. 5. Pressure test for two saturation conditions during diversion gel injection. Maximum gel strength G*=7.5 Pa at 10 rad/s and 90 ºC. ................................... 201 Figure 5. 6. Parallel cores test with two permeabilities. Core 1, high permeability K=216 mD; Core 2 Low permeability K=49 mD.............................................. 202 Figure 5. 7. Parallel cores test with stimulation. Formulation maximum gel strength (G* at 10 rad/s) of 9.8 Pa. Left: flow rates. Right: Mobility ................................ 203 Figure 5. 8. Above: Permeability restoration for all single core tests. Below: Productivity recovery for all single core tests ............................................................................. 207 Figure 5. 9. Left: Pressure test at a flow rate of 4 mL/min, after three days of injection. ....................................................................................................................................................... 210 Figure 5. 10. Rheological behavior of a formulation at 149 ºC at different dilutions ratios with diesel .................................................................................................................. 211

1 December 2015

INTRODUCTION

Increasing oil well productivity can be achieved by producing more than one interval in the same completion. If the intervals have different petrophysic characteristics like different pressure, permeability, formation damage or water invasion, well intervention will be necessary to treat a particular zone within the completion. Depending on the type of treatment and completion configuration, it is sometimes mandatory to isolate a specific productive interval to divert a chemical treatment to the desired zone. This could be done in order to protect the productive zone from the chemical treatment or to maximize the placement of the treatment in the desired interval. Diversion methods are used in oil industry to divert fluids to specific intervals by isolating a productive zone. A successful diversion treatment should avoid the inflow of a chemical treatment in the isolated interval and support the bottom-hole pressure induced during injection. Some diverting agents used are chemicals which are placed in the zone to be isolated before the further chemical treatment is injected. Chemical diversion can be used during a variety of treatments, like stimulation, water shutoff treatments and fluid loss in low pressure zones during drilling or cementation. This work will be focused in a proposed formulation for chemical diversion consisting in an oil-based self-degradable gel aimed to be injected as a low viscosity liquid in the zone to be isolated. This system should crosslink by a temperature induced reaction and the gel formed should resist further chemical treatments, avoiding their inflow into the isolated zone. This gel should degrade itself after some specific period of time and flow out of the oil well, returning the isolated zone to its original permeability conditions.

Introduction

2 December 2015

The developement of this formulation requires the characterization of the rheological behavior of the polymer system throughout the structural changes occurring during liquid-solid transition as a consequence of crosslinking and degradation reactions. The crosslinking process is characterized by a liquid-solid transition and the viscoelastic changes should be analyzed with precision to get information about its kinetics. Because the crosslinking process involves a chemical reaction, the liquid to solid transition can be directly related to the kinetics of these reactions. The relaxation modes present in these reactions might appear at long times and the equilibrium modulus should be greater than zero. The polymer chosen for this purpose was the styrene-butadiene rubber, which in presence of an organic peroxide and temperature is capable of crosslink by covalent bonds through a radical reaction. The polymer network is transformed into a self-degradable gel system by incorporating two different radical initiators. The first one has a short decomposition time and acts as a radical initiator for the crosslinking reaction. The second one, with a longer half-life time, decomposes itself slower and acts as a degradation agent by promoting β-scission reactions in the polymer network leading to low molecular weight soluble chains. The competition between crosslinking and degradation reactions results in a self-degradable gel. During liquid-solid transition, structural changes occur in the system affecting molecular mobility, leading to large changes in rheological behavior, which can be studied by small amplitude oscillatory shear experiments taking into account the contribution of structural conformation of the network at different scales, even if only a small fraction of the spectrum is actually sampled. For practical reasons, strain solicitation is most frequently used for dynamic experiments, because the experimental times are shorter and the viscoelastic behavior can be evaluated under steady conditions, such as the linear range. Dynamic experiments involve a wavy stimulus over the sample, as a sinusoidal variation of the

Introduction

3 December 2015

strain for the case of relaxation tests. Measuring the response of the material leads to characteristic viscoelastic behavior. Probably the most commonly used model to describe the viscoelastic behavior of materials is derived from a mechanistic idealization of materials known as the Maxwell model, which includes the relaxation time. The relaxation time spectrum is a useful function that can be calculated from experimental data . From dynamic rheology, most of the material functions can be calculated without any ambiguity and may also be linked to kinetic theories. The relaxation spectrum reveals characteristics of viscoelastic behavior that may not be revealed by plots of storage and loss moduli, hence, several authors recommend to use spectra to study the polymer or material viscoelasticity. Nevertheless, relaxation time spectrum cannot be directly measured, and its calculation involves the resolution of an ill-posed problem. Hence, there is not a unique solution and small changes in experimental data are traduced in large changes in the relaxation time spectrum. Several methods have been developed to calculate the relaxation time spectrum from viscoelastic experimental data and most of them are briefly explained in Chapter I. One of the major problems encountered in small angle dynamic rheology for these kinds of systems is that the data is only available in a limited frequency range. Therefore, the development of a method capable of calculating the relaxation time spectrum from experimental data with a short range of frequency is important for applications where time-temperature superposition is not possible, i.e., crosslinking and degradation reactions that cannot be stopped. In Chapter II, a new method is presented to calculate a continuous relaxation time spectrum from short frequency range experimental data, which can also be used to describe complex systems. In Chapter III, the rheological behavior of the SBR self-degradable gels is evaluated by small angle oscillatory in-situ and ex-situ experiment and the relaxation time spectra were calculated with this new method. During the liquid-solid transition the

Introduction

4 December 2015

relaxation spectra reveals the different relaxation modes related to the structural changes of the polymer system. Because the system under study involves the crosslinking and degradation reactions occurring simultaneously, and the chemical changes were not concentrated enough for a chemical characterization, for the study of the degradation, a different approach was used. The SBR in solution was submitted to degradation using hydroperoxide in anaerobic conditions. In Chapter IV the kinetics aspects of this degradation are evaluated. In this case, the fragmentation rates of polymer chains were calculated from the change in molecular weight distribution with time. The characterization was performed with a multiangle light scattering coupled with a size exclusion chromatography (SEC-MALS). The used of population balance equations was important to calculate some kinetics constants and from the analysis of the results it was possible to determine the main scission mechanism of the SBR degradation with hydroperoxide. None of these results will determine the applicability of the SBR self-degradable gel system. The closest approach to a real field application can be only performed in physical simulations with porous media. Returning to the application of the system under study, a typical concern in diversion operations for oil wells can arise in complex treatments, where diversion time needs to be adjusted accurately. If the isolation is reversed too quickly, there is a risk of invading a productive zone and cause a damage that may affect the productivity. If on the other hand, the isolated zone takes too much time to recover its original permeability, the productivity is comprised by the deferred production time. Then, Chapter V closes the studies by evaluating different formulations of the oil-based self-reversible gel in porous media. Crosslinking time, maximum strength and degradation time are adjusted by controlling the crosslinking and degradation reaction kinetics at a specific temperature.

Introduction

5 December 2015

This doctoral thesis is presented as a compilation of articles published or to be published in specialized indexed journals, each publication corresponding to a chapter.

6 December 2015

CHAPTER I

THE RELAXATION TIME SPECTRUM

FROM RELAXATION DATA: A

THEORETICAL REVIEW

Oscar Vernáez, A.J Müller

Doctoral student contribution: In this chapter, the student compiled, reviewed and wrote all the information regarding the methods found in literature for the calculation of the relaxation time spectra. The review will be submitted to an indexed journal in the near future with the student as correspondence author.

Chapter I: The Relaxation Time Spectrum From Relaxation Data: A Theoretical Review

7 December 2015

ABSTRACT

This study covers a complete revision of methods reported to calculate discrete and continuous relaxation time spectra from relaxation experimental data of viscoelastic materials. Approximation methods, analytical methods, minimization, regularization methods and some other recently proposed methods are summarized and explained. Calculation of relaxation time spectrum depends on how the method proposes to deal with the ill-posedness arising from the Fredholm integral. Because it is a ill-posed problem, there should not be a unique solution, but recently advances on the subject point to finding of a relaxation time spectrum to be continuous, smooth, positive and between a reasonably range of relaxation time which will represent more completely the viscoelastic behavior of materials. The validation and comparison of methods are briefly mentioned based on previous works, but this study has no intention to rank or prioritize any method over the others. The aim is to show the evolution and the availability of methods for calculation of the relaxation time spectrum from experimental data.


8 December 2015

RESUMEN

Este estudio abarca una completa revisión de métodos reportados para calcular los espectros de tiempos de relajación, tanto continuos como discretos, a partir de datos experimentales de materiales viscoelásticos. Se resumen y explican los métodos de aproximación, métodos analíticos, métodos de minimización y regularización y algunos otros métodos propuestos recientemente conseguidos en la literatura. El cálculo del espectro de tiempos de relajación depende de cómo los métodos propuestos lidian con el problema mal condicionado que se deriva de la integral de Fredholm de primer tipo. Debido a que es un problema mal condicionado, no debería existir una única solución, pero avances recientes sugieren que la representación más completa del comportamiento de un material viscoelástico debería consistir en un espectro contínuo, suave, positivo y dentro de un rango razonable de tiempos de relajación. La validación y comparación de métodos se menciona brevemente fundamentados en trabajos previos, aunque es importante acotar que este estudio no tiene la intensión de jerarquizar o priorizar ningún método sobre el resto. El objetivo es mostrar la evolución y la disponibilidad de diferentes métodos para el cálculo del espectro de tiempos de relajación a partir de datos experimentales.


9 December 2015

RÉSUMÉ

Cette étude est une revue exhaustive des méthodes utilisées pour calculer des spectres de temps de relaxation, aussi bien discrets que continus, à partir de données expérimentales de relaxation sur des matériaux viscoélastiques. Les méthodes d’approximation, les méthodes analytiques, les méthodes de minimisation, de régularisation, ainsi que d’autres méthodes proposées plus récemment sont présentées et expliquées. Le calcul du spectre des temps de relaxation dépend de la façon dont on se propose de résoudre le problème mal posé lié à l’intégrale de Fredholm. Comme c’est un problème mal posé, il ne devrait pas avoir de solution unique, mais les travaux les plus récents montrent que l’on peut représenter les propriétés viscoélastiques du matériau de la façon la plus complète en utilisant un spectre de relaxation continu, lissé, positif et se situant dans une gamme raisonnable de temps de relaxation. La validation et la comparaison des méthodes sont brièvement présentées sur la base des résultats de la littérature, mais notre propos n’est pas de classer les méthodes ou d’en sélectionner certaines. Notre objectif est plutôt de montrer l’évolution et la disponibilité des méthodes de calcul des spectres de temps de relaxation à partir des données expérimentales.


10 December 2015

Introduction

The use of the linear viscoelastic theory to characterize materials has always been an important topic in rheology, and its extension to link the rheological behavior with the molecular structure and conformation of viscoelastic materials has involved serious effort since the first half of the 20th century. From measurements of viscoelastic properties of polymers, the nature of macromolecular arrangements and interaction in short and long range can be inferred. In particular, polymeric systems relax with a broad spectrum of relaxation modes. Longer modes can be related to the motion of entire molecules or chain segments and shorter modes are characteristic for small scale motion of the polymer chain segments. When relaxation times are too long, they may be reflecting the presence of large scale structures, related with intermolecular associations and supra structures. The relations between stress, strain, and their time dependences are in general described by the constitutive equations (Ferry 1980). The concept of linear viscoelasticity is most often expressed in terms of the integral representation of the Boltzmann superposition principle where sequential changes in strain are additive (Ferry 1980): ')'()'()( dttttGt

t

∞−

−= γσ (1.1)In a typically viscoelastic material the time necessary for the material rearrangements to take place is comparable with the time scale of the experiment. The concept of relaxation comes from the empirical observation of relaxation experiments where each material needs a characteristic time to relax stress after some strain is applied. This concept evolved due to the need of linking this behavior with the physical conformation of viscoelastic materials. Hence, the existence of a distribution of relaxation time was introduced.

Chapter I: The Relaxation Time Spectrum from Relaxation Data: A Theoretical Review

11 December 2015

The linear theory of incompressible viscoelastic materials has been thoroughly elaborated. The linear relaxation modulus plays an important role in many constitute equations, and there has always be desirable to have an analytical representation of G(t) in term of physical conformation of polymer chains, like Rouse model (Rouse Jr 1953, Ferry et al. 1955), the Doi-Edwards (1988) theory of entangled systems, reptation (De Gennes 1979) and double-reptation models (Des Cloizeaux 1990, Leonardi et al. 2000), among others. From these studies one can conclude that there should be only one function which characterizes a viscoelastic material. Probably the most commonly used model to describe the viscoelastic behavior of materials is derived from a mechanistic idealization of materials known as the Maxwell model, which include the relaxation time. All linear viscoelasticity theories are based on this model or its variations. A function which is needed to be inferring from experimental data is the relaxation time spectrum. From this, most of the material function should be calculated without any ambiguity and may also be a link to kinetic theories. The relaxation spectrum reveals characteristics of viscoelastic behavior that may not be revealed by plots of storage and loss moduli, hence, several authors recommend to use spectra to study the polymer or material viscoelasticity. From constitutive equations is possible to represent the relaxation modulus as an integral function of the relaxation time spectrum. However, the relaxation time spectrum is not a physical property which can be measured directly from any experimental configuration. Its calculation involves a Fredholm integral of the first kind, which inversion represents an ill-posed problem. This mean that there is not a unique solution for the relaxation spectrum, beside that small changes in experimental data are traduced in huge changes for the relaxation time spectrum. Since, many different methods have been proposed in the literature to calculate this relaxation time spectrum.


12 December 2015

Many efforts concerning the resolution of ill-posed problems in general can be found in the mathematical or physical literature. In the scope of the present study, the focus lay on those method presented in rheology related works. Winter and Mours (1997) have indicated that “the choice of an algorithm for determining a discrete relaxation spectrum is a personal preference rather than objective definition”. On the same direction Tschoegl (1993) pointed that “No line spectrum is ever the true spectrum. This necessarily remains unknown. Furthermore, no such spectrum is unique because its exact nature depends on the parameters introduce to obtain it”. There have been several main works for the study of the viscoelastic properties of materials and the relaxation time spectrum. For this study, major contribution was found in the works of (Ferry 1980, Tschoegl, Emri 1993, Winter, Mours 1997), The Honerkamp’s research group (Honerkamp, Weese 1989, Honerkamp 1989, Elster, Honerkamp 1991, Elster et al. 1992, Elster, Honerkamp 1992, Honerkamp, Weese 1993, Friedrich et al. 1996, Thimm et al. 1999, Roths et al. 2000, Roths et al. 2001) with a huge contribution to the resolution of the ill-posed problem with regularization method; while for the numerical methods and mathematical tools, the works of Hansen (1994, 2007) have provided important information as well for this review. On the other side, interconversion between relaxation moduli and creep compliance is assumed to be possible and then only this study will be focused only on relaxation time spectrum and not retardation spectrum. Interconversion methods can be study from several other contribution (Ferry 1980, Mead 1994, Anderssen et al. 2008). There are also works where methods for the calculation of retardation spectrum are presented (Kaschta, Schwarzl 1994, Dooling et al. 1997), but these studies will remain out of the scope of the present review.


13 December 2015

Being obvious that relaxation time spectra studies have been extensively carried out to cover some developments in the material characterization field, and an endless list of applications can be found in literature, some studies are important to be highlight. Winter’s and Schausberger’s group (Winter, Chambon 1986, Chambon, Winter 1987, Baumgaertel, Winter 1989, Baumgaertel et al. 1990, Jackson et al. 1994, Jackson, Winter 1996, Winter, Mours 1997, Carri, Winter 1997, Nandi, Winter 2005, Friedrich et al. 2008, Brabec, Schausberger 1995) have developed their models for calculation of relaxation time spectra from empirical observations. Their studies on crosslinked system and liquid-solid transition, as well as the rheological characterization of polymer systems with well defined molecular weights have provided some crucial information for the development and applications of relaxation time spectra. Other important more recently researches where relaxation time spectrum is the main characterization parameter can be found in literature: (Stadler et al. 2008, Resch et al. 2011, Kontogiorgos et al. 2009, Friedrich et al. 2008, Friedrich et al. 2009, Nobile, Cocchini 2008, Bhattacharjee et al. 2012, Vernáez, Müller 2014, Vernáez et al. 2015) A recent discussion over the applications of relaxation time spectrum for characterization of viscoelastic materials is presented by Stadler (2013) Maxwell models for viscoelastic materials

The form of the time dependence of the relaxation modulus G(t) or the frequency dependence G’(ω) and G’’(ω), can be modeled by the generalized Maxwell model, which consist in an arrangement of springs and dashpots. For viscoelastic solids, one of the relaxation times must be infinite as a spring without dashpot with an equilibrium modulus Ge. Any stress relaxation process which decreases monotonically can be fitted with a series of terms gi.e-t/ti, where gi correspond to value of the strength of each spring and the ti the relaxation time relate to each Maxwell element. The function gi(τi)


14 December 2015

correspond to the discrete relaxation spectrum. If the number of Maxwell elements is increased without limit, a continuous relaxation spectrum arise H(τ). it

n

iie

te egGdeHGtG ττ ττ /

1

/ ln)()( −

=

∞

∞−

− +=+= (1.2)

For a oscillatory (sinusoidal) strain deformation applied on a viscoelastic material, the dynamic response is related with the stress diphase and is express in terms of the storage and loss modulus G’(ω) and G’’(ω), which are the decomposition of the viscoelastic response in phase and out of phase. The relaxation modulus can be express in terms of the dynamic moduli G’(ω) and G’’(ω) as follows (Schwarzl 1971): ωωωωω

πlncos)(''sin)('1)( dtGtGtG

∞

∞−

+= (1.3)Then, from equation (2) the dynamic modulus can be express in terms of the Maxwell models and the relaxation time spectrum as:

++=

+

+= =

∞

∞−22

22

122

22

1)(ln

1)()('

i

ii

n

iiee gGdHGG

τωτωττ

τωτωτω

(1.4)

+=

+=

=

∞

∞−22

122 1

)(ln1

)()(''i

ii

n

iigdHG

τωωτττ

τωωττω

(1.5)The G’(ω) is defined as the stress in phase with the strain in a oscillatory shear deformation divided by the strain, and is called the storage modulus because it measures the energy stored and recovered on each deformation cycle. Frequency can be approximate to the inverse of the time, hence, G(t)≈G’(1/t) (Ferry 1980). The terms in brackets are often referred as the Kernel function of the Maxwell model.


15 December 2015

Since G’ and G’’ are directly related to G(t), they must also be related to each other thorough the known Kramers-Kronig Relationship(Dealy, Larson 2006): ∞

−−=

0222

/)(''/)(''2)(' dxx

GxxGGω

ωωπω

ω

(1.6)

∞

−−=

022

)(')('2)('' dxx

GxGGω

ωπω

ω

(1.7)

At low frequencies, G’(ω) becomes proportional to ω2 and it would be independent of molecular weight distribution but the proportionality constant G’ and G’’ can be also expressed as a function of the relaxation modulus G(t) through the following constitutive equations (Winter, Chambon 1986)

∞

=0

)sin()()(' dtttGG ωωω (1.8)∞

=0

)cos()()('' dtttGG ωωω (1.9)From the experimental point of view, representation of relaxation moduli in terms of the frequency has many advantages. Almost all methodology for the calculation of relaxation time spectrum are based from one way to the other to the representations of Maxwell models presented above.


16 December 2015

Relaxation time spectrum

One of the first terms introduced to express the relation between the experiment time scale and the response time of the material was the Deborah number, which approaches to zero for a pure viscous material and goes to infinity for pure elastic materials. This represents a mean value for all contribution at molecular scale of the material. Considering that macromolecular systems may be based on properties which may differ in the material structure, as molecular weight distribution, multiphase behavior or thermocomplex systems, the representation of the relaxation arising from all these properties with just one scalar magnitude may hide the structural contributions. The relaxation time spectrum represents the contribution in strength of every material characteristic behavior which may be model as a Maxwell element, associate with their correspond relaxation time. The magnitude and shape of the spectrum reflect the material structure. Its calculation implies the resolution of an Fredholm integral of the first kind and hence, the solution of this kind of integral equations is mathematically an ill-posed problem, because small variations in the function G(t), or G’ can cause large oscillations in H(τ)(Tschoegl 1989). The response of the material over a stimulus may lead to very different relaxation times and the range may include several orders of magnitude. Hence, the relaxation time spectrum is often express in the logarithmic scale, and in most of the method to be describe, they are presented equally spaced. For any positive integer n<N, it can be assumed that log(τn+1/τn)=Δlogτ>0 where Δlogτ is a constant. A logarithmic distributed relaxation time spectrum can be represented as:


17 December 2015

;log1

1log1τ

ττ N

N −=Δ

(1.10) τττ log)1(loglog 1 Δ−+= nn (1.11)

If a continuous and unique spectrum exists, the following condition may applied for the solution algorithm (Cho, Park 2013) • The algorithm should result in a non-negative spectrum • The spectrum should fit the modulus data quite well. The norm of the vector ||Gexp-Kgi|| should be acceptably small. • The shape of the continuous spectrum should be independent on the number of relaxation times. • The solution spectrum must be stable for the errors in data.

Hence, the main question arised how to determine the spectra from experimental data for some material function. Methods for inferring the relaxation time spectrum from data start by obtaining a discrete spectrum that meets certain stringent criteria to overcome ill-posedness. Then, methods are applied to obtain continuous spectrum from the discrete one. It is important to evaluate the method to see whether a resulting spectrum is a property of the sample rather than an artifact of the procedure. Nevertheless, as it was stated before, there is no such thing as a unique spectrum, and the interpretation are linked to the experimental conditions and the nature of the samples.


18 December 2015

Ill-posed problems

An ill-posed problem are those problem where there is not an unique solution or if it is not a continuous function of the data, i.e, small perturbation of the data can cause an arbitrarily large perturbation of the solution(Hansen 1994). Inverse operators associated with equations (1.2),(1.4)and (1.5) are not continuous. It can be assumed that obtaining a function that cannot be measured directly, an inverse problem arises, and in most cases it is ill-posed(Friedrich et al. 1996). For any iteration method, absolute convergence may not be possible if data is contaminated by some errors because it is an ill-posed problem(Cho 2013). The exponential term in equation equations (1.2), and the term in brackets in equations (1.4)and (1.5) are called Kernel functions K and can be expressed as a N×M matrix where N is the number of frequencies and M is the number of relaxation times. Errors in data of G’ and G’’ may amplify the errors in the spectrum due to some small singular values of this matrix K (Honerkamp, Weese 1989). Hence, analytical methods have been proposed to overcome ill-posedness by approximation of the spectrum function to one which facilitates the analytical resolution of the Fredholm integral, either by discretization or by variable transformation as Laplace or Fourier. On the other side, iterative methods include regularization terms to minimize a cost function depending of the spectrum and some prior knowledge of the spectrum. Both discrete and continuous solutions have been proposed and there will be discussed in the following sections.


19 December 2015

Approximation methods

First approximation method The first approximation method depend upon replacing the Kernel function of (1.2), e-

t/τ by zero or unity accordingly as t/τ > 1 or < 1, or in equation (1.4)and (1.5) on replacing the fraction ω2t2/(1+ω2τ2) and 1/(1+ω2 τ 2) by unity or zero depending wheter ωt < 1 or ωt >1 (Ferry, Williams 1952). If H(lnτ) is the spectrum, then the first approximation method leads to (Schwarzl, Staverman 1953):

τ=

−=≅ttd

tdGtHtHln

)()(ln)(ln 1 (1.12) Which has been called the Alfrey’s rule (Ferry 1980). Identically can be defined from the storage modulus as:

ωωω ln)(')/1(ln')(ln 1 ddGHtH =≅ (13) And from loss modulus, the first approximation is calculated by multiplying by certain normalization vector:

)(''2)/1(ln'')(ln 0 ωπ

ωτ GHH =≅ (14) Replacing G(t), G’(ω) and G’’(ω) in these equations with Ge = 0 leads to:

τττ

τ ln)(ln)(ln /1 dHettH t

∞

∞−

−= (15)


20 December 2015

( ) τττωτωω ln)(ln

12)/1(' 222

22

1 dHH ∞

∞− +=

(16) ττ

τωωτ

πω ln)(ln

12)/1('' 220 dHH

∞

∞− +=

(17) The functions inside integral, coming along with H(lnτ) function, are called intensity functions. The narrower the intensity function is, with the limit case of the Delta function, the better would be the approximation. Nevertheless, these approximations are very poor because they can only yield the overall shape of the spectra. Higher derivative methods From the first derivative method, Schwarzl and Stavermann (1953) showed how approximations of increasing order can be calculate from the relaxation modulus by making use of inverted Laplace transform.

+−−=

−−=

≅

=

=

τ

τ

33

3

2

2

3

22

2

2

ln)(

21

ln)(

23

ln)()(ln

ln)(

ln)()(ln

)(ln

t

t

tdtGd

tdtGd

tdtdGtH

tdtGd

tdtdGtH

tH

(1.18)

The approximation of n-th order involves the logarithmic derivatives of the experimental function of the nth order and these approximations tend to the exact spectrum as the order of derivative increase because the intensity functions become narrower. From dynamic moduli the calculation turns complicated because inversion problem during transformation. Schwarzl and Stavermann (1953) proposed to settled a condition where transformation of ωτ to 1/ωτ remains invariable, then, f(lnωτ)=f(-

lnωτ). Thus, for the real part of dynamic modulus, an approximation of first order


21 December 2015

exists but second order approximation does not exist. The next higher approximation does exist and is expressed in the form: 3

3

3 ln)('

41

ln)(')(ln'

ωω

ωω

dGd

ddGtH −= (1.19)

From the imaginary part, no approximation of the first order exists but the second’s expressed as:

−= 2

2

2 ln)(')(''2)/1(ln''

ωωω

πω

dGdGH (1.20)

The preceding approximations normally require graphical or numerical differentiation of data. Tschoegl (1989) presented additionally a series of approximations based on finite differentiation by meaning of difference operators, which transform the intensity functions in linear combinations and some spectral shift factors appear to locate the peaks in the spectral space. For example the first approximation obtain from the relaxation modulus, τβ

τ=

−=

thhtGtGH

ln)()(

1 (1.21) where h is a positive integer and β=ln h/ (h-1). Tschoegl (1989) showed higher order approximation based on finite differentiation. (Ninomiya, Ferry 1959) derived from a linear combination of the Kernel functions of G’(ω) and G’’(ω), including a constant in the corresponded intensity functions and they finally expressed the H’1 and H’’2. The need of low order approximations results in an underestimation of the height of the peak of a spectrum but has a relatively minor effect on the slope (Tschoegl 1989). In general, the spectra should be a smooth function which slowly changes as a time depended response of the material, then, differentiation methods for the estimation of


22 December 2015

the spectral functions seems to work well. A recent work (Bhattacharjee et al. 2012) has successfully used this approximation method for asphalt characterization. Power law approximation Ferry and Williams(1952) derived approximation to the spectra based on a power law relation between the spectrum and the relaxation modulus. Assuming that relaxation spectrum is a decreasing function of τ, the relaxation spectrum can be presented over the range of interest by a power law expression which takes the form:

m

HH−

≅

00)(

τττ (1.22)

Then, -m is the slope of a double logratihmic plot of H against τ. Replacing this approximation in the relaxation modulus equation and changing variable τ=t/x the following is obtained: )()(

00

0

1

00 mHGdxexHGtG

m

exm

m

e Γ

+=

+=

−∞−−

−

ττ

ττ (1.23)

where Γ(m) is the gamma function. Derivation of equation (1.23) and replacing in first approximation in equation (1.12) leads to the following approximation m

HmH−

+Γ=

001 )1()(

τττ (1.24)

Then,


23 December 2015

)()(log

)(log()()()( 1 τττ

HmMtdtGdtGmMH

t

=−== (1.25)

where M(m)=1/Γ(m+1). Equation (1.25) should be a better approximation for the spectrum than the first approximation. Matrix decomposition Method Even though this method is not normally taken as reference, we considered it is a precursor for methodologies different from approximation methods and introduce the idea of regularization, error and matrix inversion, all method that will be presented in further sections. Clauser and Knauss (1968) proposed to solve Fredholm integral introducing a constraint which minimized the curvature of the solution. They also proposed the representation of the continuous part of equation (1.2) as matrix decomposition from Simpson quadrature:

=

+=N

ii

1jjiji εGHKW

(1.26)

with j>i ; Hi=H(τι), Gj are the vector of experimental values of the relaxation moduli, Kij=log10K(ω,τ), being K the kernel function, Wi is a vector with the Simpson quadrature coefficients and εj are the relative error associate to each measurement. Then, defining a matrix Aij=WiKij, and keeping the number of relaxation modes low to avoid the ill-posedness, with a pseudo-solution presented as:

j1

iji GAH −= (1.27) They proposed to calculate the second order difference of Hi to measure the curvature of the spectrum, and minimize a function composed of the square error with respect


24 December 2015

to the number of Maxwell elements and the curvature. They even include a constant which they defined as a smoothing factor but it was actually a regularization parameter. This constant weights the contribution of the minimization of the curvature in comparison with the error. They also applied the power law approximation to calculate the spectrum from the relaxation modulus. The contribution of Stanislav et at. (1974) is related basically to the introduction of the precursor of the conditional number is defined from the maximum and minimum eigenvalues of the kernel matrix. The ratio of the maximum eigenvalues and the miminum determines the lower limit of the maximum possible error in the spectrum. this ratio also suggests the degree to which the kernel function simulates the properties of the Dirac delta function. However, they only present the numerical method for determining this ratio for spectrum calculated from approximation methods and from Clauser and Knauss. Laplace method Laplace methods or analytical inversion had been used in early works (Albrecht, Freudenthal 1961) dedicated to calculated discrete relaxation time spectrum from Maxwell model. These authors started from the shear rate equation as a function of stress and the discrete components of the relaxation spectrum. They solve analytically the transform of the strain for three Maxwell elements and presented some relaxation time spectrum calculated from experimental data using this method. They warned about the selection of relaxation times, as by spacing the elements too far from each other lead to deviation of the results. Their spectra were calculated in a relaxation time range over forty decades. Other authors (Friedrich et al. 1995) had represented the dynamic moduli in parametric expressions called fractional modes, derived from fraction standard solid models. Then, these fraction modes can be derived by applying inverse Fourier transform to obtaind a generalized Mittag-Leffler function. An analytical expression of the relaxation time spectrum is obtained by applying the Laplace transform to this


25 December 2015

function, but it implies that the relaxation modulus for zero time and at the equilibrium time had to be known. Few authors had recently present relaxation time spectrum calculated from this way, one example could be (Tan et al. 2000). Padé-Laplace Method Several authors (Simhambhatla, Leonov 1993, Fulchiron et al. 1993, Carrot, Verney 1996) had presented an improved method to calculate relaxation time spectrum by using mathematical tools as Taylor series, Laplace transform and the Padé approximant. The method involves three main steps. First the Laplace transform of the relaxation modulus and some derivates at some particular point p0 are evaluated through numerical integration of the equation G(t) as a function of G’ and G’’ using experimental data. Second, this Laplace transform is represented through Padé approximants that has to be built from the previous Taylor coefficients. Third, the Padé approximants provide the discrete relaxation spectrum through their poles and residues. The method does not require any assumption concerning the number of modes becuase it is given by the method itself. The methodology is briefly presented in more detail. Starting from the relaxation modulus in its discrete form as equation (1.2),

ii tn

ii

tn

ii egegtG μτ

=

−

=

==1

/

1)( (1.28)

Since relaxation times are real and positive, μi=-1/τi. Then the Laplace transform of the relaxation modulus equation in terms of μi can be expressed in its continuous and discrete form as follows:


26 December 2015

=

∞−

−==

n

i i

ipt

pgdttGepGL

10

)()(μ (1.29)

where p is a complex number. The values μi=p are poles of function, but the Laplace transform converges only in a region without the poles. Then, to evaluate the transform in the whole complex plane, the Taylor series S of the Laplace transform function is evaluated in the vicinity of a point p0 chosen in the convergence plane of the right-half of the complex plane.

r

rr

r

rr

r

r ppr

r

pcppcpdp

pGLdr

S ')(')(!

10

000

0

∞

=

∞

=

∞

= =

=−=

= (1.30)

with p’=p-p0. The cr are called the Taylor coefficients defined as:

0

)(!

1

ppr

r

r dppGLd

rc

=

= (1.31) Now, the Padé approximation consists in writing a rational polynomial fraction of the function LG(p) in terms of the Padé approximants.

( )( )120

10

1

00

00 1)(

)()( +

=

−

=∞

=

−++−

−=−

N

N

N

r

rr pp

ppb

ppappc θ

β

ββ

α

αα

(1.32)

The coefficients aα and bβ are the Padé coefficients which can be obtained by solving the system of linear equations following from this equation. In the ideal situation, θ(p-

p0)=0. Then, equation (1.32) becomes:


27 December 2015

( ) ∏

∏

= =

=

−

=−∞

= −−=

−=

−−

−−=−=

n

i

n

i i

i

i

in

n

n

nr

rr p

gp

g

Bppb

AppappcpGL

1 1

10

1

101

00 /1)(

)()()(

τμβ

β

αα

(1.33)

The roots B of the denominator can be calculated and they are the poles of the Padé approximation function. As long as N is greater than the number of relaxation modes, the poles of the Padé approximants corresponding to the relaxation times will stabilize, thus the number of relaxation modes can be determined from G(t). The method can be extended to obtain the spectrum from the storage and the loss modulus using the equation (1.3), with its Laplace transform continuous and discrete expressed as: )()(')(''1ln)(')(''1)( 2222 peW

pGpGd

pGpGpGL

ii

i

iii +++=

++=

∞

∞− ωωωω

πω

ωωωω

π (1.34) The method provides a complete mathematical approach and seems to determine the number of relaxation time because it is based on classic mathematics tools. However, the procedure suffers from trial and errors because the number of relaxation time is determined whenever the order of Taylor expansion is larger than two times the number of the relaxation modes. The order of the Taylor expansion of the Laplace transform of the relaxation modulus must be increased until the number of poles determined by the Padé-Laplace does not change (Cho 2010). The relaxation time spectra obtained method depends whether p0 or the center of the Taylor expansion of the Laplace transform has been correctly chosen.


28 December 2015

Emri-Tschoegl Algorithm This method (Emri, Tschoegl 1993, Emri, Tschoegl 1995) consists of predetermining a set of responses times which are equally spaced on the logarithmic time axis, then calculate the N-th strength of the spectral line associated with each relaxation time within a fixed time interval or “window” around this relaxation time and taking away the point which. The algorithm is based on the fact that the kernel functions can be presented as step functions. From the continuous relaxation modulus, if the kernel function at some relaxation time exp(-t/τκ) is plotted against ln t/τκ , a first window is chosen where this kernel shows the largest time dependence. Identically, the derivative of the kernel function at τ=τκ−1, τκ, τκ+1, are plotted as a function of ln t/τκ, then the intersection of the central derivative function with its neighbors functions define a second window which has to be closer than the first window. There should be at least one discrete data point within this last window. In the method, they select some datum points from the relaxation modulus using a windows methodology to evaluate relaxation times and their vicinities. They split the discrete relaxation modulus, in three parts around some k relaxation time and included a term to account the experimental error in the source data:

jt

Ni

kii

tk

tki

iiej

ijkjij egegegGtG ετττ ++++= −=

+=

−−−=

= /

1

//1

1)( (1.35)

Two decades to the right of the location of any τi, the value of the exponential can be neglected. Inside a selected window, the method finds the values of gi that minimize the sum of squares of εj, which leads to:


29 December 2015

=

−

−

=

Φ+= sn

sj

t

tsn

sjje

kkj

ij

e

etGg

1

/2

/

1)(

τ

τ

(1.36)where,

ijkjij tNi

kii

tk

tki

iiejj egegegGtGt τττ /

1

//1

1)()( −

=

+=

−−−=

= +−−−=Φ (1.37)

Starting from the longest time or the N-th term, where the sum terms vanished, it is possible to calculate through successive steps the gk terms and so the relaxation time spectrum. The method has some inconvenient when negative spectrum are generated and strongly depends on the windows methodology and its generator steps. The authors also applied the methodology to calculate the relaxation time spectrum from storage and loss modulus (Emri, Tschoegl 1993) and from other rheological experimental data. Winter-Chambom spectrum Winter and Chambon (Chambon, Winter 1987, Winter, Chambon 1986) evaluated polymer systems which undergo in liquid-solid transition, especially those crosslinked by covalent bonds. They identify the gel point through dynamic oscillatory tests observing, for the case of Polydimethylsiloxane systems, that in the gel point, G’(ω) and G’’(ω) coincide in a broad frequency range. Using the temperature shift factor, they might reproduce a broad frequency range. Their experimental work had led to some conclusions regarding the relaxation time spectrum at the liquid-solid transition related to the power law behavior.


30 December 2015

At low frequencies their samples exhibit either liquid behavior, with G’ proportional to ω2 and G’’ proportional to ω, or solid behavior, where G’ remains constant and G’’ appear proportional to ω. At high frequency, this power law will remain if there is no glass transition involve. Then, to make G’ and G’’ equals and proportional to ω1/2 (in order to agree with Kraemers Kronig realation of equation (1.6), the relaxation modulus should be

nFttG −=)( (1.38)Where F is a characteristic “stiffness” of the material and n is restricted to values between 0 and 1. In order to put this equation as the conventional way to represent relaxation spectrum H(τ) needs to be:

n

nFH −

Γ= )(

)()( ττ (1.39)

It follows that for this particular experiment where G’ and G’’ are equals in a broad range of frequency, the solution of the relaxation time spectrum has a coincidence with the Rouse model. This model is then called CW-spectrum. The longest relaxation time is defined by the following equation: n

e

FG /1

max

−

=τ (1.40)

An extensive discussion over the use of CW-spectrum in liquid-solid transition has been presented by Winter and Mours (Winter, Mours 1997).


31 December 2015

Minimization methods:

Until this point, most of the methods presented, are based essentially in approximations or assumptions. Despite most of the previous methods cannot be considered as obsolete and is out of the scope of this publication to make any judgment about which method is better, there is a history line from where methods start to demand some validation against experimental data. From this line, what almost all method have in common is how the calculated relaxation time spectrum, reproduced the original data after a simple conversion from constitutive equations. Then, it is possible to use a cost function which evaluate the moduli generated by the calculated spectrum, and compare them with the experimental data. The minimization methods start from this cost function, and applied different methodologies to calculate the spectrum by iteration until the cost function is minimized. If the residual is low, data generated from the calculated spectrum should reproduce the input experimental data. Cost Functions Depending on the method to be used, these cost functions can be presented in several forms which may be essentially the same. Baumgaertel and Winter (1989), presented a method which uses the minimization of the following equation to determine the best spectrum. This cost function evaluates simultaneously the generated storage and loss moduli. Cost function is generally presented as a least square curve fitting problem.

( ) min1)('

),,('1

)(''),,(''

,2

exp1

2

exp

=

−+

−=Ψ

= j

iijm

j j

iijii G

gGG

gGg

ωτω

ωτω

τ (1.41) Some authors as Cho(2010, 2013) prefers to use only one modulus to be included in the minimization algorithm, because it allows simplification of the method. They normally compare the results after finalization of the algorithm.


32 December 2015

( ) dwwgwGgw

w

N

iiiii

2

1

max

min

)()('',

−−=Ψ=

λφλ (1.42) ( ) [ ]

= ==

+−=−=Ψ

m

j

n

i ij

ijj

m

jiijjii giGgGGg

1

2

122

22

exp1

2exp1 1

)('),,(')(',τω

τωωτωωτ (1.43)

( ) [ ] = ==

+−=−=Ψ

m

j

n

i ij

ijj

m

jiijjii giGgGGg

1

2

122exp

1

2exp2 1

)(''),,('')('',τω

τωωτωωτ (1.44)

The mathematical manipulation of cost function may lead to linear problems that may be normalized. For example, Cho (2013), presents the minimization of ψ1 as the normal equation: ij

m

jnjni

i

n

iniiij

KKQ

gQgG

=

=

=

=

1

1),,(' τω (1.45)

Or expressed in vectorial and matricial form leads to: TKKQ

QgG== (1.46)

Least square minimization problem can be solved by Levenberg-Marquardt algorithm, for linear and non-linear systems. Nevertheless, the spectrum calculated by this method generally leads to negative values, noisy or totally random strength points of the spectrum without any physical meaning. Other authors as Orbey and Dealy (1991) and Vernáez and Müller (2014) use the standard deviation as cost function, including Ge, expressed as follows:


33 December 2015

=

==

+−+

−

+−=Ψ

m

j j

n

i ij

iji

ej

n

i ij

iji

G

g

GG

g

m 1

2

)exp(

12

2

)exp(

12

22

'')(1

.1

')(1

.11 τω

τωτω

τω

σ

(1.47)

Honerkamp and Weese (1989) proposed a cost function that includes the standard errors of each measurement εj as follow

= =

−=Ψm

j

n

ijiij

j

G1

2

1)exp(2 .'1

2 Kgεχ (1.48)

The next methods are based from one way to another, in minimization of cost functions. Parsimonious discrete spectrum IRIS® This method (Baumgaertel, Winter 1989, Baumgaertel et al. 1990) is based on nonlinear regression avoiding ill-posedness by diminishing the number of relaxation modes. Details of the methods are contained in a commercially available Interactive Rheological Software System “IRIS®” They start by combining equations (1.2) and (1.22) they expressed the continuous spectrum in discrete form as:

itn

i

m

eHtG τ

ττ /

1 00)( −

=

−

= (1.49)

with spacing α of the relaxation times, i.e. τi=τo10iα, where τo is the shortest mode in power law spectrum. The parameters of a discrete relaxation spectrum are found by fitting G’ and G’’ equations to the experimental Gexp’ and Gexp’’ data, which means the minimization of equation (1.41) through least-square minimization.


34 December 2015

Differentiation of this cost function with respect to gi leads to a set of linear equations which can be solved numerically. When the space between the relaxation times are too small, some negative coefficients may appear, that is gi<0. In this particular method, both gi and ti are freely adjustable, as well as the number of modes n which is a crucial consideration in the method. Too few relaxation modes leads to a large residual, but these residual decays rapidly when increasing the number of relaxation modes, until certain value is reached, from which residual changes become negligible. The number n is also keep small enough to avoid ill-posedness. The initial number of relaxation modes is chosen empirically between 1 and 2 per decade, then the number of relaxation modes is increased progressively until the optimum number is achieved, where the residual is reduced to a value from which changes are not significant and ill-posedness arises. Hence, the solution should be unique for small number of relaxation modes. This method does not assume equally spaced times, and the number of relaxation times and relaxation strengths are all optimized iteratively to provide the best fit of data. The representation of this discrete spectrum is called Parsimonious spectrum. The spacings between neighboring modes depend on the specific material. The authors assume that there always exists a discrete set of Maxwell modes which represents the continuous spectrum accurately (Baumgaertel, Winter 1992). They demonstrate that for any continuous spectrum, a representative discrete Parsimonious spectrum should exist, and then they postulated that any Parsimonious spectrum can be converted into a continuous spectrum. BSW Spectrum In the same way that Winter and Chambon studied relaxation time spectrum directly on experimental data, Baumgaertel et al. (1990) presented an empirical study using Parsimonious spectrum to evaluate polymers of different molecular weights.


35 December 2015

They observed that the short modes for all samples overlaid on a single straight line with negative slope and long modes differentiate in straight lines with positive slopes. Then they proposed to decompose the spectrum in two contributions, the shorter modes were related to the glass transition, which is independent of molecular weight and the flow regime for the longer relaxation modes. Because of the linear behavior it was evident that both short modes and long modes spectra follow a power law fit. The power law spectrum with negative exponent was then associated with the Chambon-Winter spectrum. Curiously, the calculation allowed the two regimes to overlap in the time scale. In flow regime they assume an ideal spectrum of the form: )/1()/()( maxmax τττττ −= hnGH n

N (1.50)being GN the equilibrium modulus (when G’ tend to infinity), τmax the longest relaxation time, and h(x) a unit step function. The spectrum of this characteristic was then called BSW-spectrum which is and incomplete gamma function. On the other side, the CW spectrum for the glass regime is represented as equation (1.39) which has a characteristic lowest relaxation time on the high frequency limit. The BSW and CW spectra have been compared with the Rouse model (1953) and Doi-Edwards (1988) models for physical association. For the Rouse model describes the terminal zone very well, but it continues directly into the glassy region without allowing plateau modulus at intermediate frequencies, which is typical for linear flexible monodisperse polymers of low molecular weight, near the critical molecular weight of entanglement Mc. Winter (1997) has published a summary of the IRIS algorithm highlighting the main features of the program and some characteristic of the method. Some interesting applications of these spectra are for example, the characterization of linear Polystyrene (Schausberger 1991); for nearly monodisperse polybutadiene


36 December 2015

melts (Baumgaertel, Winter 1992); afterwards Jackson et al. (1994) compiled the extended applications for linear monodisperse polymers. Studies for critical gels as presented by De Rosa and Winter (1994) and Winter and Mours (1997). Regularization methods From the work of Clauser and Knauss (1968) the use of a condition for the resulting spectrum as a constraint is implemented. In that case, they constraint the solution of the spectrum with a curvature, and they minimize a function including the curvature. Thus, regularization refers to the inclusion of additional constraint to solve an ill-posed problem. This constraint can be perceived as a penalty for complexity, such the restrictions of smoothness or bounds. A linear problem can be expressed in matrix form as Ax=b. If this equation corresponds to an ill-posed problem, the least squares approach leads to an overdetermined or underdetermined system of equations. In least-squares method, the objective is to minimize the sum of squared residuals, expressed as ||Ax-b||2, where ||.|| is the Euclidean norm. This expression is the equivalent to the cost function expressed in previous section. Then, a new term can be add to this minimization leading to:

22 xx ΩbA +− (1.51) The last term is added to include another minimization condition, the same way as Lagrange multipliers imposed an equality constraint. This term is called regularization term. The most common regularization is the Tikhonov regularization (Tikhonov, Arsenin 1977), where the solution of the problem is the weighted combination of the residual norm and the side constraint,

22

222

xxx ΩbA λ+−= (1.52)


37 December 2015

Thus, regularization can be included in the cost function, and then instead of minimize the function Ψ(gi,τi), a new cost function expressed as ΨR(gi,τi) =Ψ(gi,τi) + λ2||Ω||, should be minimize, where Ω is a model operator. The value λ is called the regularization parameter which weights the contribution of the regularization term to the final solution. In many cases, the operator is chosen to be the identity matrix (Ω=I), giving preference to solutions with smaller norms. Some authors (Clauser, Knauss 1968, Honerkamp, Weese 1989, Orbey, Dealy 1991) use the regularization operator Ω can be an operator which transforms the vector gi into its second derivative, indicating the smoothness of the vector. Then, minimizing ΨR(gi,τi) means finding the vector gi which best fit the experimental data, but also is smooth enough because its second derivative is also minimized. When the regularization parameter is too high, minimization prioritizes the smoothness over the fitting with experimental data, and when the regularization parameter tends to zero, the solution gi is found such that moduli generated from it has the minimum deviation from experimental data, but no concern over the smoothness of gi is taken. Tikhonov regularization presents difficulties in the determination of relaxation time spectra because the considerably different contributions at short and long relaxation times. From a mathematical point of view, this is a peculiar property of the relaxation spectrum (Honerkamp, Weese 1993) Hansen (2007)present certain regularization tools include in Matlab library which can be easily used for solving ill-posed problems, and other authors as Kontogiorgos (2010), has cited the mentioned Matlab tools to calculate relaxation spectra.


38 December 2015

Singular value decomposition The singular value decomposition (SVD) is referred as the factorization of a complex matrix. It is based on the possibility of represent a matrix A ∈Rm×n in its factorized form:

=

==n

i

Tiii

T vwu1

U.W.VA (1.53) Where U=(u1,…,un) ∈ Rm×n is a matrix composed of n columns and ui eigenvectors of length m, which are orthonormal to the matrix A.AT. The matrix W∈ Rm×n is a diagonal matrix with non-negative real values, named singular values, and V=(v1,…,vn) is a matrix composed of n columns and vi are orthonormal eigenvectors of the matrix A.AT. The singular values of the matrix A are represented in the diagonal of the W matrix, where the scalars (w1, …wm) are non-negative values and appears in decreasing order, wi≥wi+1. Because matrices U and V are orthonormal, they satisfy UT.U=VT.V=I. Matrix SVD allows to calculate the pseudoinverse of the matrix A+. Let A=U.W.VT, the pseudoinverse matrix of A, can be written as A+=V.W+.UT, where W+ is the matrix form by replacing all elements in the diagonal of W by their reciprocals (1/w1, …,1/wm). From the pseudoinverse it is possible to find an approximated solution to a least-square problem including ill-posed. Discrete ill-posed problems in linear systems such as Ax=b, A∈Rm×n and linear least-squares problems

,min 2bAxX

− A∈Rm×n, m>n. (1.54)


39 December 2015

The singular values of A decay gradually to zero and the ratio between the largest and the smallest nonzero singular values is large (Hansen 1994). If the spectrum can be calculated from equation (46) then a pseudo-solution as proposed by Clauser and Knauss can be only obtained if the number of Maxwell modes are keep in the same dimension as the number of frequencies, in order to obtain a matrix A (or S in equation (46) ) invertible. Using SVD equation (1.27) can be expressed as:

=

++ ===m

ij

Tii

j

T vw1

).(1.* GuGUV.WGAH jjiji (1.55) The condition number is a way to quantify how ill-posed a problem is. When the condition number is one, then a solution algorithm can be found, but if the condition number tends to infinity it implies that the problem is ill posed. In term of the singular values, the condition number is defined as the ratio between the maximal and minimal singular values of the matrix A which is also the ratio between the maximal and minimal eigenvalues of A. The Tikhonov regularization method can be applied if the regularization term is included for the singular value decomposition. For this purpose the general singular value decomposition (GSVD) can be used (Honerkamp, Weese 1989, Orbey, Dealy 1991, Roths et al. 2001, Hansen 1994) and the Aij matrix can be expressed as:

1

1

).0,.(

.0

0.

−

−

−

=Ω

=

XMV

XI

WUA

pnij

(1.56)


40 December 2015

Where Ω ∈ Rp×n is the regularization operator, which in the case of Tikhonov regularization it returns the second derivative. U ∈ Rm×n and V ∈ Rp×p, are orthonormal matrices; X∈ Rp×p, and W and M ∈ Rp×p are diagonal matrices formed by the elements (w1,w2…wp) and (ε1,ε2…εp) and satisfy: (0≤w1≤w2…≤wp≤1); (1≥ε1≥ε2…≥εp≥0); such that (wi2+εi2)=1. The cost function can be expressed then in terms of the GSVD as: 2

Ω

−

=Ψ

λλi

iij

GSVD

GH

K (1.57) The calculated spectrum is then presented as:

( ) iT

ijT

ijT

iji GK.KKH ...1−

ΩΩ+= (1.58) Maximum entropy method The maximum entropy method has been widely used for other applications and is base in the definition of entropy S of a given distribution H (τ).

∞

∞− ≥

−≅−=1

ln)(ln)()(i

ii ggdHHHS τττ (1.59)Using entropy as the regularization operator Ω, leads to the following cost function (Hansen 1994, Elster, Honerkamp 1991):

≥= =

+

−=Ψ1

2

1

2

1)exp(2 )log(.'1

2

iiii

m

j

n

ijiij

j

gpgG λεχ Kg (1.60)


41 December 2015

where pi are weights of each gi to the entropy function. The use of this particular function arise from the fact that entropy can be written from a prior known distribution H0(τ) (or the distribution from the previous iteration) as the following equation(Elster, Honerkamp 1991): ∞

∞−

−= ττττ ln

)()(ln)()(

0

dHHHHS (1.61)

Taking the discrete form being g0 the previous iteration value of the discrete vector (Hansen 2008): 0

10ln)( ii

N

i i

ii gg

gggHS +−

−=

= (1.62) Maximizing S means to gi=gi0, and implies a regularization of the cost function. CONTIN algorithm Provencher (1982) presented an algorithm named CONTIN used for inverting noisy linear operator, including Fredholm integral equations of the first kind which can be used for minimization problems with constraints and regularization. This algorithm is often used for calculation of relaxation time spectrum based on Thikonov regularization. Writing the constitutive equations (1.2), (1.4)and (1.5) in the generic form with an specific relaxation time interval [a,b], valid for relaxation, storage and loss moduli, and including some error the following equation is obtained:

ετττ += dxKHxGb

a

),()()( (1.63) where K(x) is the Kernel function which depend on the modulus chosen or the model.


42 December 2015

CONTIN algorithm creates a discrete form of (1.63) k

N

mkmkmk ετ +=

=1)( gKG (1.64)

Then the algorithm admits some constraint for the solution gk as non-negative values and the regularization is included in the minimization cost function as follow: ( ) kkkmkii Mg RgrgKG −+−=Ψ 22

)(, λτ ε (1.65) In the algorithm the regularization term included r and R, which are user selected and penalizes the solution gk for deviations from behavior expected on the basis of statistical prior knowledge or parsimony. In this sense, parsimony is a strategy to select the solution smooth and with the minimum number of peaks, which in our terms would be the second derivative, as it was said before. The method, on which the CONTIN algorithm is based, is the singular value decomposition that will be explained in further sections and the method for choosing the regularization parameter is based on the parsimony, i.e , taking the largest regularization parameter that is consistent with the data. Non-linear regularization methods (NLREG) Honerkamp and Weese (1993) show that Tikhonov regularization performs very poorly at long relaxation times. Then, instead of inferring H(τ) from data, their algorithm search for log(H(τ))= ξ(τ)., which involves a nonlinear regression. The use of a regularization parameter enables the selection of the best fit given the experimental error and prevents over-fitting the data to a level beyond their accuracy. The method tracks noise propagation and revelas clearly its effect on the spectrum produced. In this method (Honerkamp, Weese 1993), a new cost function is defined as:


43 December 2015

( )

∞

∞−=

∞

∞−

=

∞

∞−

+

+−−

+−=Ψ

ττ

τξλττω

τωω

ε

ττω

τωω

ετ

τξ

τξ

ln)(ln101

)(''''

1

ln101

)(''1,

2

2

22

1

2

)(22exp

1

2

)(22

22

exp

dd

ddG

dGg

m

j j

jj

j

m

j j

jj

jiiNLR (1.66)

In their work, the minimization is performed by a program NLREG, developed by the authors, but the details are not presented. Another regularization applied for rheology problems is the edge preserving regularization (EPR) method (Roths et al. 2000) where the relaxation time spectrum is considered as a composition of several smooth spectral zones instead of a unique spectrum. The EPR method solves spectra with sharp edges or large curvatures, which may be the case of BSW and CW spectra. By changing the regularization operator, the method constraints the solution to one with a similar shape to the BSW and CW spectra, with smooth change in the boundaries. The regularization term is presented as: [ ]

ττ

ττ

τ

τln

)(''.11.)(''ln

.1

1.22

2

22

2

dHa

Hd

dgda

dgd

i

iEPR

+=

+

=Ω ∞

∞−

∞

∞− (1.67)

where a is a positive arbitrary constant which imposed smoothness to the spectrum. The solutions for these regularizations are summarized in the work of Roths et al. (2001) where they presented an algorithm for solving generalized regularization method, including non-linear and edge preserving regularization. In this work, the regularization is considered a generic operator of the solution, i.e, is a function of the solution to be found.


44 December 2015

KKT minimization Vernáez and Müller (2014) proposed a method for minimization for problems expressed either with Tikhonov or EPR regularization, using a double constraint method based on the Karush-Kuhn-Tucker conditions, i.e, minimizing the first gradient of the cost function in equation (1.47), using a optimization software. Thus, the method minimizes the cost function with the regularization term under an additional linear constrain to make the solution closer to the real unperturbed solution as: bHKH ij −⋅=)(y (1.68)

The optimization algorithm implies the minimization of a new cost function with the sum of all constrains as: )()()()( HHHH yKKT ∇+Ω∇+Ψ∇=Ψ (1.69)

The solution obtained by the optimization routine minimizes the distance of the two surfaces formed by the cost function and the regularization method. The method is used particularly for experimental data in a short frequency range, and needs to amplify the range of the relaxation times beyond the inverse of the frequency or the range calculated by the sampling localization theorem (Davies, Anderssen 1997). This topic will be discussed in a further section. Bayesian analysis Most of the regularization methods can be considered in the Bayesian framework. In the work of Hansen (2008), a regularization with a new constraint factor which combines the conventional smoothness constraint of the second derivative and the maximum entropy constraint.


45 December 2015

The second derivative regularization operator in its discrete form, considering equally spaced relaxation times in the logarithmic scale, can be described as: 222

)(22

11

1

211 N

N

i

iiii

gggggg ++

+−=Ω

−

=

+− (1.70) where g1 and gN can be considered to be zero to make a close distribution. Then, taking for g=g0, and making an approximation to a second order Taylor series, equation (1.62) of the maximum entropy method becomes;

( )=

−≈N

i i

ii

gggHS

10

20

2)( (1.71)

Thus, using gi0=(gi-1+gi+1)/2, the maximum entropy constraint is defined as a new smoothness constraint scaling by a 2gi0 factor. Combination of the smoothness constraint and the maximum entropy metric was demonstrated to be appropriate for estimation of relaxation modulus. In this method, a probability function is provided to calculate the regularization parameter to be used in the minimization of the cost function. Bayesian method provides a large number of solutions with associated probabilities for each set of data, which makes it possible to calculate reliable errors for the average relaxation modulus. The users do not have to provide any additional input; it calculates the regularization parameter by itself. It also quantifies the number of good independent parameters. Choosing regularization parameter Since many of the above methods include a regularization parameter, they all need to set a criterion for optimize this parameter. As it was said before, large values of the


46 December 2015

regularization parameter will give a higher contribution to the constraint imposed in the regularization term and hence, the difference between experimental data and calculated from the obtained spectrum is compromised. Each estimate of the regularization parameter will correspond to an estimate of the noise level of the experiment as the cost function is minimized. From the works presented in this study, it is possible to pick some methodologies applied to calculate the optimum regularization parameter. Hansen (1994) summarized the methods for choosing the regularization parameter, identifying four methods: discrepancy principle method, L-curve method, the generalized cross-validation and the quasi-optimality criterion. The L-curve method is mostly used for the general singular value decomposition applied to a Tikhonov regularization method (Kontogiorgos et al. 2009). Has been useIn L-curve is a graphical method from a log-log plot of the norm of the regularization term versus the corresponding residual norm calculated for several regularization parameters, and the optimal is chosen from the L-curve’s corner. A modified version of the L-curve method was implemented by Vernáez and Müller (2014), where instead of picking the L-curve’s corner, the norm of the vector formed by ||Kij.g-Gexp|| and ||Ω(gi)|| normalized from 0 to 1, is calculated for each value of the regularization parameter. Then, the optimum regularization parameter is search by minimizing the norm: norm

inormijoptimum g )(min 2exp Ω+−= G.gK

Hλ (1.72)

The self-consistent algorithm proposed by Roths et al. (2001)implemented in programs as NLREG, minimizes the mean deviation of the estimated spectrum from the true function. In this method, the regularization operator is applied to a function of the prior iteration spectrum as a function of the regularization parameter.


47 December 2015

Honerkamp and Weese (1989), as well as Orbey and Dealy (1991), proposed to use a method where the regularization parameter is chosen from the minimization of: 2

1 1min

= =

−=

N

i

M

jiij

i

ioptimum gg λ

σλ K

H (1.73) In the Bayesian method (Hansen 2008) a probability is conferred to each estimate of the relaxation spectrum related to the regularization parameter, making possible to calculate an average spectrum from the set of all spectra having non-negligible probabilities. It also gives a probability distribution for the regularization parameter. The probability P for the regularization parameter can be calculated according to:

( ))2/(det

)2/exp(121)( 2/1

2/1

Ψ∇∇+Ω∇∇Ψ−Ω−

+

= λλ NP

N (1.74) Simulated annealing Iterative methods can be applied to minimize the cost function as the one presented in equation (1.47). Jensen (2002), proposed the minimization using a Monte Carlo method called simulated annealing. This method is based on the idea of the thermal annealing phenomena, where the minimal energy state of a material is obtained. The cooling rate determines the molecular rearrangement and the final energetic state. Thus, a slow cooling rate allows the rearrangement until a global minimum is achieved, while an abrupt cooling rate freezes the molecules to storing energy. From this principle, an algorithm is designed to minimize or optimize a cost function, equivalent to energy, throughout another evaluation function to find a global minimum. It consists in searching a local minimum through gradual perturbations of the solution until some local minimum is obtained. Then, the method performs the following iteration which is allowed to be worse than the previous, in order to escape from local minimum. The algorithm starts with a high “temperature” defined with


48 December 2015

high initial values, and then the first optimization process is performed until some stopping criterion is reached. Then, the next iteration step is considered as a decrease in temperature, escaping from local minimum. As the temperature descends, the iteration step to escape from local minimum is less abrupt, until the optimization stay in the global minimum. For a given temperature, the random values of gi are determined as follows: )().()(* qFqRqgg ii += (1.75)

gi* is the value to be evaluate, gi(q) is the optimum value of the previous iteration q-1, R(q) is a random number from -1 to 1; and F(q) is the parameter which indicate how large would be the escaping step from the local minimum. For each gi* a cost function is optimized, then if the residual of gi*(q) are smaller than the gi*(q-1), it would stay as reference global minimum, otherwise, the solution jump randomly. To control random movements from solutions, a probability function is used as the search progress, making movements or jumps smaller for every iteration, or temperature. Normally, an exponential decay function is used as a function of temperature:

Ψ−Ψ

= Tgg

i

ii

egP*)()(

)( (1.76) Then, Ψ(gi)-Ψ(gi*), which is < 0, represents a new perturbation of the solution. Note that as the temperature decreases, the probability function decreases as well. For simulated annealing method, the convergence to a global solution has been probed for slow cooling rates, which implies longer calculation times.


49 December 2015

Jensen also changed the number of Maxwell relaxation modes the same way as parsimonius method, along with the simulating annealing to optimize the calculation time. The method was evaluated with different relaxation time ranges. Linearization Method The cost functions are linear functions of the moduli and non-linear by relaxation times. Malkin and Kuznetsov (2000) proposed to modify the problem of the relaxation time estimation to transform the problem in linear by factors depending on τi. The relaxation times are found from the linearized approximation and then the partial moduli are found by minimization of the cost function, by meaning of least-root-square method. The moduli function are presented as a polynomial power series

)()()(' 2

2

ωωω

n

n

FD

G = (1.77) )()(1)(' 2

2

ωω

ωω

n

n

FZ

G = (1.78)where Fn(x), Dn(x) and Zn(x) are n-order polynomials:

1...)1()( 21

121

20

1

222 +++=+= −−

=∏ ωωωωτω n

nnn

iin fffF (1.79)

0...)1(..)( 21

121

20

2

,1

222

1

2 +++=+= −−

≠==∏ ωωωωτωτω n

nnn

ijjii

n

iin dddgD (1.80)

0...)1(..)( 21

121

20

2

,1

22

1

2 +++=+= −−

≠==∏ ωωωωτωτω n

nnn

ijjii

n

iin zzzgZ (1.81)


50 December 2015

With this transform it is possible to use linear method for minimization, as least-root-squares, for instead of finding directly the gi,τi pairs, the values of the polynomial coefficients. The method has several issues concerning the calculation of coefficients for storage and loss modulus independently, which leads to two different solutions, and also, there is not a constraint regarding the shape or physical meaning of the obtained spectrum. Finally, the method does not avoid the ill-posedness and after some polynomial order, or Maxwell modes, the results start to show waviness. Smoothness factor Kaschta and Stadler (2009) proposed a method to avoid waviness by constraining the minimization with a smoothness factor which can be derived from the shape of the relaxation spectrum itself assuming a smooth preferable decreasing spectrum. The smoothness factor concept is introduced and it is related to the change in the double logarithmic slope between subsequent relaxation modes and it is expressed in terms of the least square error. The relaxation times are chosen equidistantly spaced on the logarithmic scale. Letting pd be the number of modes per decade, its inverse can be express as:

di

i

p1log 1 =+

ττ (1.82)

The double logarithmic slope is expressed as: )log(log

loglog

1 iid ggpd

Hd −≈ +τ (1.83)

Then the smoothness factor is defined as:


51 December 2015

( )[ ]−

=++ −−−=

1

1

411 loglogloglog

n

iiiiid ggggpSF (1.84)

Then in highly noisy spectrum, the slope between subsequent relaxation modes changes from negative to positive values, but positive slopes contribute twice as much as the negative slopes. This smoothness factor is added to the cost function in the optimization routine the same way as regularization terms are include, hence, can be considered a regularization method. In this methodology, they assume that the relaxation spectrum is a decreasing function in order to create favorable conditions in the optimization routine. Cubic Hermite Spline Stadler and Bailly (2009) have proposed the Cubic Hermite spline method to calculate a continuous spectrum (piecewise cubic Hermite interpolating polynomial). In this method, some spectrum descriptors are defined, called knots, which are feely adjustable in the routine. Initialization of the method is performed by calculating spectrum with the method of the smoothness factor (Kaschta, Stadler 2009), but the spectrum descriptors do not have to be equidistant. Then this spectrum is normalized using Baumgartel and Winter (1992) transformation equation. Then, the relaxation times are fixed and the relaxation spectrum is optimized by the best cubic Hermite spline which describes the continuous relaxation spectrum. Since Cubic Hermites are polynomials, computation is simple. The optimization is performed using a cost function. After this first optimization, the spectrum descriptors are treated as freely adjustable parameters in both, relaxation time and strength to optimize the resulting spectrum. Finally, the spectrum is adjusted to get rid of those parts of the resulting spectrum that do not have influence or are just artifact and to make it physically meaningful.


52 December 2015

The highest and lowest frequencies are selected accordingly to sampling localization theory of Davies and Anderssen (1997). The optimization is based on the “pchip” routine and the Hermite spline is performed with the “interp” routine, both in Matlab™, The routine starts with a small number of knots, 3 by default. If the cost function is reduced by adding one extra knots, the new spectrum is taken to the next loop until certain user-defined number of knots is achieved. The method deal with random error in data by smoothing them before inferring a spectrum They use interpolation to fit a piecewise function of cubic Hermite splines to the spectrum. There is also a penalty to the error calculation for each decade where the slope is larger than a certain value that is considered unrealistic (McDougall et al. 2014). Wavelet Transform (WT) method Cho (2010) introduced a method to determine the number of relaxation times and identify them without any adjustable parameters or prior assumption. The WT method is based on continuous wavelet transform (WT) and on an interesting approach for the decoupling of peaks. It minimizes a cost function of equation (1.42), considering only the loss modulus. The wavelet transform is an integral transform that converts the loss modulus into a function of relaxation time in logarithmic scale. This function is composed of peaks whose maxima can be approximated to the values of the gi and are centered at the logarithm of the inverse of relaxation time (-log τi). The loss modulus can be express in terms of a relaxation time and frequency in the logarithmic scale:


53 December 2015

−==

N

iiwgiwG

1)()('' λφ (1.85)

where λi=-logτi and w=log ω, and the function: x

x

ee

xx 21cosh2

1)(+

==φ (1.86) which is a symmetric peak. Thus, φ(w-λi) is a symemetric peak shift by λi. Minimization of the cost function defined in terms only of the loss modulus in equation (1.42) is achieved searching both gi and λi by the least-square method with the conditions:

0=∂Ψ∂=

∂Ψ∂

iig λ (1.87) The minimization is expressed as two linear matricial problems, using a square invertible matrix of the Kernel function φi, which may lead to non-positive values of gi. Since, function φi is a symmetric peak, the inner product <φ(w- λi),φ(w- λj)> will be nearly zero whenever the difference τi-τj becomes considerably large and it will be a finite value whenever τi-τj approaches to zero. If the spacing between relaxation times on logarithmic scale is sufficiently large, a series of peaks appears whose maxima are located at τj. They use a wavelet to transform the G’’ peak in to sharpener one, but still centered in λk. The problem is that the relaxation times have to be sufficiently spaced or the gi values will have a severe deviation. If relaxation times are closely distributed, a wavy wavelet transform is obtained. Additionally, the values of G’ and G’’ calculated from


54 December 2015

the spectrum obtained deviate from the experimental data at low frequencies and, for validation test the method does not reproduce well a given known spectrum. One advantage is that WT method determines the number of relaxation times without any additional procedure. The resolution of WT is limited to the width of half maximum of the wavelet transform of φi(w), which is about less than unity in the logarithmic scale of frequency. The method requires a refinement because of the interference between Maxwell modes. Fixed point iteration method The method of fixed point iteration (Cho, Park 2013) was developed to calculate the continuous spectrum of linear viscoelasticity. As the WT method, the minimization is performed over a cost function with only with loss modulus, which leads to the normal equations (1.45) or (1.46). The authors suggest an iterative algorithm based on Landweber iterative regularization which can avoid the occurrence of negative values of g.

=

+ += m

j

rjnj

rn

rn

g

Ggg

1

)(

)()1( ''logloglogQ

, for 1≤n≤N (1.88) or,

=

+ = m

j

rjnj

rn

rn

g

Ggg

1

)(

)()1( ''

Q (1.89)

This is called a fixed-point iteration. The fixed point is not unique when the matrix Q is not invertible.


55 December 2015

The matrix Qnj is approximated as follows: jnnj

njnjQ

ττττττ

ω //)/log(

log1

−Δ≈ (1.90)

where Δlogω=log(ωi+1/ωi). This approximation is valid for a logarithmic spacing of frequency of less than unity. Differently from the WT approximation, in this case, the approximation is broader than the original kernel. The matrix Q does not have the peak-like shape, which is the reason why the storage modulus is not used as data. The successive approximations are based on probability density functions which analytically approximate the r-th iteration as a function of the previous iteration. If the total number of frequencies N is less than the number of relaxation times, the iteration algorithm must not have a unique fixed point. An arbitrary initial spectrum is transformed iteratively to give a spectrum that progressively improves the fit of the data. After some number of iteration, the algorithm converges to a solution or fixed point. The number of iterations is established by trial and error to identify an optimum. The method does not depend on initial spectrum, because it will be cancel in equation (1.89). The iteration makes initial spectrum close to the original spectrum rapidly when the curvature of the original spectrum is small. The sensibility of the method to data errors was demonstrated using singular value decomposition and comparing the eigenvalues and the largest singular value. The algorithm seems to be stable regardless of the presence of errors in modulus data, despite they did not formally tested it with a specific deviation.


56 December 2015

Power series iterative method

Cho(2013) uses the advantage of the variable change w=log ω, , and λi=-logτi, as variables to treat analytically the constitutive equations as presented in the following equation succession: x

x

w

w

eexK

eewKK 2

2

)(2

)(2

1)('

1)(')log,(log'

+=

+=+= +

+

λ

λ

λτω (1.91)xxww ee

xKee

wKK −+−+ +==

+=+= 1)(''1)('')log,(log'' )()( λλλτω (1.92)

( ) [ ]222

2

)(''21

2)(' xKee

dxxdK

x

x

=+

= (93) Then, constitutive equations for storage and loss modulus expressed in terms of w and x are:

( )

( ) ( ) ( ) dxxKwxHwdee

HwG

dxxKwxHwde

eHwG

ww

w

w

∞

∞−

∞

∞−+−+

∞

∞−

∞

∞−+

+

−=++

=

−=++

=

)('')(1)()(''

)(')(1

)()(' )(2

)(2

λλ

λλ

λλ

λ

λ

(1.94) (1.95) Expanding G’(logw)=G’(w) and G’’(w) by a Taylor series, the following approximation is obtained:

≈ =

N

n

nnwcwG

0exp)(

(1.96) Where G(w) is referred either G’(w), G’’(w) or dG’(w)/dw.


57 December 2015

On a similar way, the relaxation time spectrum can be approximated by a power series. Basically it is assumed that the spectrum is a real analytic function in the whole real line expressed as: ( )

( )

=

=

≈

≈

n

n

nn

n

n

nn

hH

hH

0

0

)(

log)(log

λλ

ωτ

(1.97)

Then, an expansion of G’(logw)=G’(w) and G’’(w) and H(logt)=H(λ) by a Taylor series is performed to approximate the functions which leads to the following relationships: dxxKxwh

rn

wgwG

dxdx

xdKxwhrn

wgndw

wdG

rnrn

r

n

n

rn

nn

rnrn

r

n

n

rn

nn

∞

∞−

−∞

= =

∞

=

∞

∞−

−∞

= =

∞

=+

−

==

−

=+=

)('')1('')(''

)(')1(')1()('

0 00

0 001

(1.98) (1.99)

dG’/dw and G’’ are peak functions and the terms in the integral corresponds to the (n-

r)th moments of K’ and K’’ respectively. Then, the terms in the integral can be calculated analytically as function of Bernoulli and Euler’s numbers. Knowing G’’ and dG’/dw from experimental data, Gexp’ and Gexp’’ can be obtained, and then from the above equations would be possible to calculate the spectrum as a linear regression problem.

∞

∞−

−

=

∞

∞−

−

=

−=

−=

dxxKxhrn

g

dxdx

xdKxhrn

g

rnn

N

nrn

rnn

N

nrn

)('')1(''

)(')1('

(1.100)


58 December 2015

(1.101) When experimental errors are included in moduli data, there exists an optimum order of polynomial, which may be obtained similar to a parsimonious method. One problem that may arise is that sometimes it is not possible to fit G’ or G’’ over the whole frequency interval, then the frequency range should be split in subintervals. Another problem is that higher order power series regression is demanded to elucidate original model spectrum. And on the other side, very high degree polynomial leads to wavy spectra (Bae, Cho 2015). BLM method Bae and Cho (2015) used a combination of Cubic B-Spline and Lavenberg-Marquardt minimization to developed a new method to overcome some of the problems observed in the power series iterative method regarding to the instability of the spectrum for high order polynomial. The BLM method uses a cubic B-spline functions to approximate the spectrum as follows:

( )

≈ +

=

2

0exp)(

M

nnnBbH λλ (1.102)

where Bn is a 4 interval cubic based function. Then, the method consists in finding the constant bn which minimize the cost function using Lavenberg-Marquardt method. It resolves the instability of the power series method and wavy spectra. Increasing partition leads to a decrease in the value of the cost function. Also, the number of iteration needs is less than the fixed point iteration method.


59 December 2015

Because the method is based on the Lavenberg-Marquardt minimization algorithm, the method is sensitive to the initial conditions. BLM method admits experimental errors about 3% and still an acceptable spectrum is obtained. Relaxation time range

One of the fundamental problems implicit in the calculation of the relaxation time spectrum is the range of relaxation time chosen. For example, Ferry(1980) affirmed that to cover a wide enough time scale to reflect the variety of molecular motions in polymeric systems, often 10 to 15 logarithmic decades. Other authors (Clauser, Knauss 1968) indicated that two decades to either side of the transition region are sufficient to declare the range of relaxation time, which was a non-based affirmation. A common affirmation within the authors is that experimental frequency range somehow should defined the relaxation time window of the relaxation spectrum. Based on mathematical time inversion, the most commonly accepted range for the relaxation time spectrum is the one which is defined by the reciprocal of the experimental frequencies. It is impossible to access the terminal low-frequency region using small angle oscillatory shear experiments (SAOS), and creep measurements are used to probe long-time behavior. The best way is to apply the principle of time-temperature superposition to expand the experimental range, but it is still limited to two or three decades of frequency. Superposition is based on the assumption that all relaxation times of the material have the same temperature dependence, but this is reasonably accurate only over certain ranges of frequency that depend on the material (McDougall et al. 2014).


60 December 2015

Tschoegl (1993) developed a filtering algorithm leaving out a part of the experimental data, which was assumed to be less relevant for estimation of a particular part of the relaxation spectrum. The correlation between points in the relaxation spectrum imposed by a smoothness constraint may increase the effective range over which the spectrum can be estimated, but outside the theoretical limitations, the estimated relaxation spectrum is mainly determined by the regularization constraint (Hansen 2008). Perhaps the most cited works related to the frequency range estimation are those related with the sampling localization theorem (Davies, Anderssen 1997, Davies, Anderssen 1998). The sample localization theory relates the relaxation time range to the experimental as follow: 21

min21

max ..ππ

ωτω−

−− ≤≤ ee i (1.103) The derivation of the theorem is based on an approximation of the discrete spectrum by an expression of sampling distributions ψ’(ω) and ψ’’(ω):

∞∞

=−=00

)('')(')(')('' ωω

ωωψωωωψ dGdGgi (1.104) The sampling distribution are approximated by a Gauss error functions, always linked to the frequency variable. Implications of this theorem is that whatever the range of relaxation time is, it will not require more than 2.73 decades of frequency to determine the values of the spectrum from the loss modulus. On the other hand, Vernáez and Müller (2014) stated that the viscoelastic behavior should be the result of the contribution of all the Maxwell elements together, where some may be neglected, depending mostly on the physical systems and not on the experimental frequency range. Jensen (2002), included the relaxation time range as


61 December 2015

an optimization parameter E and the cost function could be optimized by increasing this range, then the maximum and minimum relaxation time is determined as: 21

min21

max ..ππ

ωτωE

i

E

ee−

−− ≤≤ (1.105) The value of E=0 represents the criteria of the frequencies inverse, and E=1 represents the sampling localization theorem proposed by Davies and Anderssen. There should be an optimal value of E that minimized the cost function (Jensen 2002). The NLREG algorithm of Honerkamp and Weese (1993), may admit larger relaxation time ranges. Some authors had studied the effect of the incomplete data sets (Stadler 2010, Vernáez, Müller 2014) concluding in both cases, using two different methods that the extension of the relaxation time range above the limits of the inverse frequencies regime can lead to better spectrum calculations. Also, they observe that it was not a matter of how many decades of experimental data are truncated, but which part of the data is more relevant to reproduce the spectrum. They both concluded that it depends on the shape of the spectrum or identically on the shape of experimental data. Naturally, uncertainty increases as the experimental frequency data range decreases. Final comments

McDougall et al. (2014) presented a work where they compare the performance of several tools for calculation of the relaxation time spectra from dynamic oscillatory data and also from generated data obtained from simulated spectra. They compared the fixed point iteration method (Cho, 2010), the NLREG algorithm (Honerkamp and Weese, 1993), the cubic hermite splines (Stadler and Bailly 2009) and the Parsimonious method or IRIS (Baumgaertel, Winter 1989, Baumgaertel et al. 1990). They conclude that the IRIS method can calculate a discrete spectrum which reproduces well the data, but the continuous spectra show more details of the


62 December 2015

viscoelastic behavior of the material, being the Cubic Hermite Spline method the most accurate but it requires significantly more computation. For any iteration method, absolute convergence may not be possible if data is contaminated by some errors because it is an ill-posed problem (Cho 2013). The error level of measurements usually falls lower than 1% for high quality data under ideal circumstances (Stadler, Bailly 2009, Bae, Cho 2015). There are three commonly used methods to validate the developed method. One is by generating data from a simulated spectrum. Usually a bimodal spectrum is used. The generated data can be contaminated with some random errors to validate the versatility of the methods to handle experimental errors. Acknowledgments

The authors would like to thank to PDVSA Intevep, for funding through the PRMI-0012-03 project in the Department of Well Productivity. Also FONACIT (Science and Technology Ministry of the Bolivarian Republic of Venezuela) and Ministry of Foreign Affairs (France) for funding received through project: PCP 2011001409 (Postgraduate Cooperation Project). References

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EMRI, Igor; and TSCHOEGL, NW. Determination of Mechanical Spectra from Experimental Responses. International Journal of Solids and Structures, 1995, vol. 32, no. 6, pp. 817-826. EMRI, Igor; and TSCHOEGL, NW. Generating Line Spectra from Experimental Responses. Part I: Relaxation Modulus and Creep Compliance. Rheologica Acta, 1993, vol. 32, no. 3, pp. 311-322. FERRY, John D. Viscoelastic Properties of Polymers. Third ed. New York: John Wiley & Sons, Inc, 1980. FERRY, John D.; LANDEL, Robert F.and WILLIAMS, Malcolm L. Extensions of the Rouse Theory of Viscoelastic Properties to Undiluted Linear Polymers. Journal of Applied Physics, 1955, vol. 26, no. 4, pp. 359-362. FERRY, John D.; and WILLIAMS, Malcolm L. Second Approximation Methods for Determining the Relaxation Time Spectrum of a Viscoelastic Material. Journal of Colloid Science, 1952, vol. 7, no. 4, pp. 347-353. FRIEDRICH, Chr; BRAUN, H.and WEESE, J. Determination of Relaxation Time Spectra by Analytical Inversion using a Linear Viscoelastic Model with Fractional Derivatives. Polymer Engineering & Science, 1995, vol. 35, no. 21, pp. 1661-1669. FRIEDRICH, Christian; HONERKAMP, Josefand WEESE, Jürgen. New Ill-Posed Problems in Rheology. Rheologica Acta, 1996, vol. 35, no. 2, pp. 186-193. FRIEDRICH, Christian; LOY, Richard J.and ANDERSSEN, Robert S. Relaxation Time Spectrum Molecular Weight Distribution Relationships. Rheologica Acta, 2009, vol. 48, no. 2, pp. 151-162. FRIEDRICH, Christian; WAIZENEGGER, Wolfgangand WINTER, Horst Henning. Relaxation Patterns of Long, Linear, Flexible, Monodisperse Polymers: BSW Spectrum Revisited. Rheologica Acta, 2008, vol. 47, no. 8, pp. 909-916. FULCHIRON, R., et al. Deconvolution of Polymer Melt Stress Relaxation by the Padé–Laplace Method. Journal of Rheology (1978-Present), 1993, vol. 37, no. 1, pp. 17-34. HANSEN, Christian. Regularization Tools Version 4.0 for Matlab 7.3. Numerical Algorithms, 2007, vol. 46, no. 2, pp. 189-194. HANSEN, Per Christian. Regularization Tools: A Matlab Package for Analysis and Solution of Discrete Ill-Posed Problems. Numerical Algorithms, 1994, vol. 6, no. 1, pp. 1-35.


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HANSEN, Steen. Estimation of the Relaxation Spectrum from Dynamic Experiments using Bayesian Analysis and a New Regularization Constraint. Rheologica Acta, 2008, vol. 47, no. 2, pp. 169-178. HONERKAMP, J. Ill-Posed Problems in Rheology. Rheologica Acta, 1989, vol. 28, no. 5, pp. 363-371. HONERKAMP, J.; and WEESE, J. A Nonlinear Regularization Method for the Calculation of Relaxation Spectra. Rheologica Acta, 1993, vol. 32, no. 1, pp. 65-73. HONERKAMP, J.; and WEESE, Jürgen. Determination of the Relaxation Spectrum by a Regularization Method. Macromolecules, 1989, vol. 22, no. 11, pp. 4372-4377. JACKSON, JK; DE ROSA, MEand WINTER, HH. Molecular Weight Dependence of Relaxation Time Spectra for the Entanglement and Flow Behavior of Monodisperse Linear Flexible Polymers. Macromolecules, 1994, vol. 27, no. 9, pp. 2426-2431. JACKSON, Judith K.; and WINTER, H. Henning. The Relaxation of Linear Flexible Polymers which are Slightly Poly-Disperse. Rheologica Acta, 1996, vol. 35, no. 6, pp. 645-655. JENSEN, Erik Appel. Determination of Discrete Relaxation Spectra using Simulated Annealing. Journal of Non-Newtonian Fluid Mechanics, 2002, vol. 107, no. 1, pp. 1-11. KASCHTA, J.; and SCHWARZL, R. R. Calculation of Discrete Retardation Spectra from Creep data—I. Method. Rheologica Acta, 1994, vol. 33, no. 6, pp. 517-529. KASCHTA, Joachim; and STADLER, Florian J. Avoiding Waviness of Relaxation Spectra. Rheologica Acta, 2009, vol. 48, no. 6, pp. 709-713. KONTOGIORGOS, Vassilis. Calculation of Relaxation Spectra from Mechanical Spectra in MATLAB. Polymer Testing, 2010, vol. 29, no. 8, pp. 1021-1025. KONTOGIORGOS, Vassilis; JIANG, Binand KASAPIS, Stefan. Numerical Computation of Relaxation Spectra from Mechanical Measurements in Biopolymers. Food Research International, 2009, vol. 42, no. 1, pp. 130-136. LEONARDI, Frederic, et al. Rheological Models Based on the Double Reptation Mixing Rule: The Effects of a Polydisperse Environment. Journal of Rheology (1978-Present), 2000, vol. 44, no. 4, pp. 675-692. MALKIN, Alexander Ya; and KUZNETSOV, VV. Linearization as a Method for Determining Parameters of Relaxation Spectra. Rheologica Acta, 2000, vol. 39, no. 4, pp. 379-383.


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MCDOUGALL, Ian; ORBEY, Neseand DEALY, John M. Inferring Meaningful Relaxation Spectra from Experimental Data. Journal of Rheology (1978-Present), 2014, vol. 58, no. 3, pp. 779-797. MEAD, DW. Numerical Interconversion of Linear Viscoelastic Material Functions. Journal of Rheology (1978-Present), 1994, vol. 38, no. 6, pp. 1769-1795. NANDI, Souvik; and WINTER, H. H. Swelling Behavior of Partially Cross-Linked Polymers: A Ternary System. Macromolecules, 05/01, 2005, vol. 38, no. 10, pp. 4447-4455. ISSN 0024-9297. NINOMIYA, Kazuhiko; and FERRY, John D. Some Approximate Equations Useful in the Phenomenological Treatment of Linear Viscoelastic Data. Journal of Colloid Science, 1959, vol. 14, no. 1, pp. 36-48. NOBILE, Maria Rossella; and COCCHINI, Franco. A Generalized Relation between MWD and Relaxation Time Spectrum. Rheologica Acta, 2008, vol. 47, no. 5-6, pp. 509-519. ORBEY, Nese; and DEALY, John M. Determination of the Relaxation Spectrum from Oscillatory Shear Data. Journal of Rheology (1978-Present), 1991, vol. 35, no. 6, pp. 1035-1049. PROVENCHER, Stephen W. A Constrained Regularization Method for Inverting Data Represented by Linear Algebraic Or Integral Equations. Computer Physics Communications, 1982, vol. 27, no. 3, pp. 213-227. RESCH, Julia A.; KESSNER, Uteand STADLER, Florian J. Thermorheological Behavior of Polyethylene: A Sensitive Probe to Molecular Structure. Rheologica Acta, 2011, vol. 50, no. 5-6, pp. 559-575. ROTHS, T., et al. A Generalized Regularization Method for Nonlinear Ill-Posed Problems Enhanced for Nonlinear Regularization Terms. Computer Physics Communications, 2001, vol. 139, no. 3, pp. 279-296. ROTHS, Tobias, et al. Determination of the Relaxation Time Spectrum from Dynamic Moduli using an Edge Preserving Regularization Method. Rheologica Acta, 2000, vol. 39, no. 2, pp. 163-173. ROUSE JR, Prince E. A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers. The Journal of Chemical Physics, 1953, vol. 21, no. 7, pp. 1272-1280. SCHAUSBERGER, A. A Description of the Linear Viscoelasticity of Molten Linear Monodisperse Polystyrenes with the Aid of a Generalized Discrete Relaxation Time Spectrum. Rheologica Acta, 1991, vol. 30, no. 2, pp. 197-202.


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SCHWARZL, F.; and STAVERMAN, AJ. Higher Approximation Methods for the Relaxation Spectrum from Static and Dynamic Measurements of Visco-Elastic Materials. Applied Scientific Research, Section A, 1953, vol. 4, no. 2, pp. 127-141. SCHWARZL, FR. Numerical Calculation of Storage and Loss Modulus from Stress Relaxation Data for Linear Viscoelastic Materials. Rheologica Acta, 1971, vol. 10, no. 2, pp. 165-173. SIMHAMBHATLA, M.; and LEONOV, AI. The Extended Padé-Laplace Method for Efficient Discretization of Linear Viscoelastic Spectra. Rheologica Acta, 1993, vol. 32, no. 6, pp. 589-600. STADLER, Florian J. On the Usefulness of Rheological Spectra—a Critical Discussion. Rheologica Acta, 2013, vol. 52, no. 1, pp. 85-89. STADLER, Florian J. Effect of Incomplete Datasets on the Calculation of Continuous Relaxation Spectra from Dynamic-Mechanical Data. Rheologica Acta, 2010, vol. 49, no. 10, pp. 1041-1057. STADLER, Florian J.; and BAILLY, Christian. A New Method for the Calculation of Continuous Relaxation Spectra from Dynamic-Mechanical Data. Rheological Acta, 2009, vol. 48, pp. 33-49. STADLER, Florian J.; KASCHTA, Joachimand MÜNSTEDT, Helmut. Thermorheological Behavior of various Long-Chain Branched Polyethylenes. Macromolecules, 2008, vol. 41, no. 4, pp. 1328-1333. STANISLAV, J.; SEYER, FAand HLAVÁCEK, B. Numerical Determination of Retardation and Relaxation Spectra Optimalization of Numerical Process. Rheologica Acta, 1974, vol. 13, no. 3, pp. 602-607. TAN, H.; TAM, KCand JENKINS, RD. Relaxation Spectra and Viscoelastic Behavior of a Model Hydrophobically Modified Alkali-Soluble Emulsion (HASE) Polymer in Salt/SDS Solutions. Journal of Colloid and Interface Science, 2000, vol. 231, no. 1, pp. 52-58. THIMM, Wolfgang, et al. An Analytical Relation between Relaxation Time Spectrum and Molecular Weight Distribution. Journal of Rheology (1978-Present), 1999, vol. 43, no. 6, pp. 1663-1672. TIKHONOV, Andreĭ N.; and ARSENIN, Vasiliĭ I. Solutions of Ill-Posed Problems. Vh Winston, 1977. TSCHOEGL, Nicholas W. The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction. Springer Science & Business Media, 1989.


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TSCHOEGL, NW; and EMRI, I. Generating Line Spectra from Experimental Responses. Part II: Storage and Loss Functions. Rheologica Acta, 1993, vol. 32, no. 3, pp. 322-327. VERNÁEZ, Oscar, et al. Dynamic Rheology and Relaxation Time Spectra of Oil-Based Self-Degradable Gels. Journal of Polymer Science: Part B: Polymer Physics, accepted 03 september., 2015. VERNÁEZ, Oscar; and MÜLLER, Alejandro J. Relaxation Time Spectra from Short Frequency Range Small-Angle Dynamic Rheometry. Rheologica Acta, 2014, vol. 53, no. 5-6, pp. 385-399. WINTER, H. Henning; and CHAMBON, Francois. Analysis of Linear Viscoelasticity of a Crosslinking Polymer at the Gel Point. Journal of Rheology (1978-Present), 1986, vol. 30, no. 2, pp. 367-382. WINTER, Henning H. Analysis of Dynamic Mechanical Data: Inversion into a Relaxation Time Spectrum and Consistency Check. Journal of Non-Newtonian Fluid Mechanics, 1997, vol. 68, pp. 225-239. WINTER, Horst H.; and MOURS, Marian. Neutron spin echo spectroscopy viscoelasticity rheology Springer, 1997. Rheology of Polymers Near Liquid-Solid Transitions, pp. 165-234.


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List of Symbols

a Smoothness parameter in EPR method aα Padé coefficient bβ Padé coefficient Aα Padé coefficient Bn 4 interval cubic based function Bβ Padé coefficient cr Taylor coefficients di polynomial coefficients in linearization method Dn(x) n-order polynomial in Linearization method E optimization parameter for relaxation times range ε Residual or error F Stiffness of the material (Winter Chambon) fi polynomial coefficients in linearization method Fn(x) n-order polynomial in Linearization method φ(x) Peak function 1/2cosh x

G Relaxation modulus G’ Storage modulus G’’ Loss modulus Ge Equilibrius modulus. Gexp Experimental modulus G’exp Experimental storage modulus G’’exp Experimental loss modulus gi discrete relaxation time spectrum- strength


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Gj are the vector of experimental values of the relaxation moduli GN Equilibrium modulus (when G’ tend to infinity) Γ Gamma function H Continuous relaxation time spectrum H0 Initial value of the spectrum or previous iteration Hi Pseudo solution of the Discrete relaxation time spectrum h(x) Unit step function K Kernel function K Kernel matrix Kij log10K vector L Laplace transform λ Regularization parameter λi logarithm of the relaxation time in WT method m Slope of a double logratihmic plot of H against τ

M 1/Γ(m+1)

μ Inverse negative of the relaxation time N number of relaxation times p Some complex number pi Weights in entropy function P Probability function of the regularization parameter in Bayesian method pd number of modes per decade P(gi) Probability function in simulating annealing Q Sum of the kernels matrix r Penalties in the regularization term (CONTIN)


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R Penalties in the regularization term (CONTIN) S Taylos Series of the Laplace transform function SF smoothness factor S(H) Entropy function σ stress γ Shear rate t time T Temperature τ relaxation time τmax Longest relaxation time U Matrix of eigenvectors ui eigenvectors orthonormal to any matrix (A.AT) V matrix of orthonormal eigenvectors of any matrix (A.AT) vi eigenvectors orthonormal to any matrix (A.AT) w Log of frequency W diagonal matrix of singular values wi singular values Wi Vector with the Simpson quadrature coefficients (Clauser and Knauss) ω Frequency ωmin Minimum experimental frequency ωmax Maximum experimental frequency Ω Regularization operator x t/τ

y linear constrain in KKT method


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Ψ Cost function zi polynomial coefficients in linearization method Zn(x) n-order polynomial in Linearization method ξ(τ) Logarithm of the continuous relaxation time spectrum

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CHAPTER II

Relaxation Time Spectra from Short

Frequency Range Small Angle Dynamic

Rheometry

Oscar Vernáez, A.J Müller

Doctoral student contribution: In this chapter, the student has proposed a new method for calculation of relaxation time spectrum from short frequency range measurements. He has also programmed the entire algorithm implicit and wrote the paper which was submitted and accepted in Rheological Acta, with the student as correspondence author.

Chapter II: Relaxation Time Spectra from Short Frequency Range Small Angle Dynamic Rheometry

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ABSTRACT

In this work, the influence of the experimental frequency range over the relaxation time spectrum is studied. The relaxation time spectra were calculated from dynamic moduli, a well-known ill posed problem, using a regularization method. The method solves the ill posed problem by simultaneous minimization of the regularized standard deviation and a restriction function. The solution was validated using a simulated spectrum. Truncated moduli data generated from simulated spectra were used to evaluate the method for smaller frequency range data. Finally, experimental data of a wormlike micellar system mixed in aqueous solution with a zwitterionic copolymer were used to validate the method. It was possible to obtain relaxation time spectra from short frequency range data if the relaxation time range is allowed to be higher than the inverse of the highest and lowest experimental frequencies. These spectra can be used qualitatively to describe complex systems when no time-temperature superposition experiments are feasible.


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RESUMEN

Es este trabajo se estudió la influencia del rango de frecuencia experimental en el espectro de tiempos de relajación. El cálculo del espectro de tiempos relajación a partir de los módulos dinámicos implica resolver un problema mal condicionado, por lo que se utilizó un método alternativo de regularización. El método resuelva el problema mal condicionado mediante la minimización simultánea de la desviación estándar regularizada y una función de restricción. La solución fue validada utilizando un espectro simulado. Los datos de módulos dinámicos generados a partir del espectro simulado fueron truncados para evaluar el método para cortos rangos de frecuencia. Finalmente, se validó el método mediante el cálculo de los espectros de relajación para datos experimentales de reología dinámica para un sistema de micelas cilíndricas en solución acuosa con un copolímero zwiter-iónico. Fue posible obtener los espectros de tiempos de relajación para datos con rangos de frecuencia cortos si el rango de tiempos de relajación se expande por encima del inverso de las máximas y mínimas frecuencias experimentales. Estos espectros pueden ser utilizados cualitativamente para describir sistemas complejos donde no es posible realizar experimentos con curvas maestras de superposición tiempo-temperatura.


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RÉSUMÉ

Dans cette partie, nous avons étudié l’influence de la gamme de fréquences étudiée expérimentalement sur le spectre de temps de relaxation. Les spectres de temps de relaxation ont été calculés à partir des modules de cisaillement dynamiques, ce qui constitue un problème mal posé, pour lequel nous utilisons une méthodes de régularisation. Cette méthode résout le problème mal posé en minimisant simultanément la déviation standard régularisée et la fonction de restriction. La solution a été validée en utilisant une simulation de spectre de relaxation. Des données de modules de cisaillement tronquées ont été générées à partir des spectres simulés, afin d’évaluer la méthode pour des gammes de fréquences plus étroites. Enfin, nous avons utilisé des données expérimentales obtenues pour des systèmes de micelles de type ver, en solution aqueuse avec des copolymères zwitterioniques. Il a été possible d’obtenir des spectres de temps de relaxation à partir de données sur des gammes de fréquences étroites lorsque les gammes de temps de relaxation étaient plus larges que l’inverse des fréquences expérimentales la plus haute et la plus basse. Ces spectres peuvent être utilisés qualitativement pour décrire des systèmes complexes, lorsque l’équivalence temps-température ne peut être utilisée expérimentalement.


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Introduction

Rheological measurements have always been the most widely used method for the characterization of polymer solutions in different applications. The viscoelastic behavior of a sample is the consequence of intermolecular associations between polymer chains, molecular weight distribution and macromolecular architecture. Complex polymer systems show different relaxation behavior that might be studied through relaxation time spectra. Nevertheless, the relaxation time spectrum cannot be determined directly from experiments. Gelation processes are characterized by a liquid-solid transition and the viscoelastic changes should be analyzed with precision if some information about the kinetics is required. If the crosslinking process involves a chemical reaction, then the liquid to solid transition will be directly related to the kinetics of these reactions. Relaxation modes present in these reactions might appear at long times and the equilibrium modulus, Ge, should be greater than zero. At high concentration, rodlike micellar systems become wormlike, and entanglements between micelles give viscoelastic properties to these systems. Polymer/surfactant systems display synergistic effects in their viscoelastic properties, due to the enhanced associations (Rodriguez et al. 2011, Shashkina et al. 2005, Rojas et al. 2010, Gouveia et al. 2009, Panmai et al. 1999, Peiffer 1999, Couillet et al. 2005). Viscoelastic properties of macromolecular systems appear when intermolecular associations build supramolecular structures. One of the major problems encountered in small angle dynamic rheology for these kinds of systems is that the data is only available in a limited frequency. For polymeric fluids at low concentrations, low molecular weights polymer solutions, low surfactant concentration in rodlike micellar systems, before any association occurred, the linear viscoelastic range is found only for a short range of frequencies

Chapter II : Relaxation Time Spectra from Short Frequency Range Small Angle Dynamic Rheometry

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and some rheometers do not have enough sensibility to measure them. As the associations increase, this frequency range extends and also the linear viscoelastic range could change. To overcome this problem, the frequency range may be expanded by preparing a mastercurve, based on the time-temperature superposition principle. Mastercurves has been used widely for dynamic rheology in many polymeric systems. Nevertheless, there are some polymeric systems whose behavior depends on temperature. In those systems, variation in temperature can lead to irreversible changes in the system. That is the case of polymer solutions associated with surfactants whose critical micellar concentration depends directly on temperature, or polymer crosslinking processes, in which reaction kinetics depend on temperature. Thermorheological complexity of some samples are also a major problem to perform a time-temperature shift, as it was demonstrated by Woods-Adams and Costeux (2001) and Stadler et al. (2008), for long-chain branched metallocene catalyzed Polyethylene. Heating or cooling the samples of some viscoelastic surfactants may change the contour length of micelles, thus the viscosity of the samples changes because of the reduction of the shortening or enlargement of the micelles for the same concentration (Shashkina et al. 2005) For liquid-solid transitions of crosslinking polymer systems, temperatures determine the rate of reaction and usually a reaction cooler is needed to stop it at a specific time or conversion to make the rheological measurements. This could also be a problem for radical crosslinking reactions, where it is not easy to stop the reaction at a specific time. Thus, the time-temperature superposition principle cannot be always applied if the temperature affects the system. Nevertheless, rheological characterization might be still performed and the viscoelastic response of these systems should be a consequence of the intermolecular association. Viscoelastic behavior of materials in the linear viscoelastic range can be represented by an array of Maxwell elements leading to constitutive equations. Stress relaxation


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experiments show how the material behaves when a constant deformation is applied to it. The stress required to support such deformation decays exponentially in time as the material relaxes, along with viscous dissipation of energy. The relaxation modulus G(t) is an exponentially decay function of the current time t and all past times t’ expressed in terms of the generalized Maxwell model as (Ferry, 1980):

=

−−

=−n

i

tt

iieHttG

1

)'(

)'( λ (2.1)where each Hi represents the elastic modulus of the i-th Maxwell element with the corresponding relaxation time λi, which is the ratio of the viscous constant and the elastic constant of each element. In the linear viscoelastic range, the relationship between stress, τ(t), and deformation, γ(t), is described by the constitutive equation of the first Fredholm integral (Ferry, 1980) as:

∞−

−=t

dtdtdttGt '

')'()( γτ (2.2)

Stress relaxation may occur fast or slow for liquid systems and there are instances where the stress could even never fully relax, e.g., for crosslinked polymeric systems. Thus, one of the characterization methods to measure the stress relaxation is small angle dynamic rheology, where a sinusoidal deformation at different frequencies is applied to the sample and the stresses required to generate such deformations are measured. From the diphase angle between the deformation and the stress, the elastic or storage modulus G’(ω) and the viscous or loss modulus G’’(ω) can be obtained. In terms of equation (2.1), the dynamic moduli can be written as a function of the relaxation time spectrum as follows:


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++=

=2

22

1 ).(1.

)('ij

ijn

iij HGeG

λωλω

ω (2.3)

+=

=2

1 ).(1.

)(''ij

ijn

iij HG

λωλω

ω (2.4))(lim tGG

te ∞→= (2.5)

where ωj are the experimental frequencies, λi are the relaxation times, n is the number of Maxwell elements and Ge the equilibrium modulus. For viscoelastic solids, stress relaxation cannot be fully achieved and this behavior can be modeled as if one Maxwell element became a Hookean element with a constant Ge. Thus, the relaxation modulus decays asymptotically to Ge with increasing time. In terms of the generalized Maxwell model, the relaxation time spectrum could be described as a distribution of the times required for Maxwell elements to relax after a deformation has been applied to the sample, e.g., a distribution of Hi with their corresponding λi. The relaxation time spectrum cannot be obtained directly from the oscillatory experiments and its calculation implies solving an ill posed problem, e.g., it has several solutions or there are many different arranges of Hi leading to the same relaxation behavior. Ill posed problems in dynamic rheology are usually linked to an inversion problem in equations (2.3) and (2.4), which means that small perturbations in the dynamic moduli (G’(ω) and G’’(ω)) would be magnified in larger variations of the relaxation time spectrum, while large changes in the spectrum do not necessarily imply big changes in the calculation of the dynamic moduli. Fitting m experimental data points, G’exp(ω) and G’’exp (ω), to the model with n Maxwell elements, means finding values of Hi and λi that generate G’ and G’’ from equations


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(2.3) and (2.4) which fit the experimental data. The fitting can be evaluated using a cost function, as the normalized standard deviation, σ, which can be written as (Orbey and Neal, 1991):

=

==

+−+

−

+−=

m

j j

n

i ij

iji

ej

n

i ij

iji

G

H

GG

H

m 1

2

)exp(

12

2

)exp(

12

22

'').(1

..

1'

).(1.

.11 λω

λωλω

λω

σ (2.6) Calculating the relaxation time spectrum is basically finding a vector H=(Hi, λi,) ∈Rn that minimizes equation (2.6). In those cases where the cost function is too flat without any outstanding global minimum, there will be more than one vector (Hi, λi,) ∈Rn, that can be very different to each other, that minimizes the cost function similarly. Then, all those vectors can be considered as solutions within a tolerance; although their physical meaning might not be appropriate. Using conventional minimization methods for solving equation (2.6), such as Levenberg-Marquardt linear regression or Newtonian differentiation, could lead to discrete spectra showing negative values with no consistence sequence between Hi elements. Even if a global minimum can be reached, it comes with a lack of physical meaning, mainly because a model for a macroscopic response of collective contribution of relaxation elements at a molecular level should be represented as a continuous distribution. For linear systems of the form:

mxnx R , ∈= AbA (2.7)The minimum square function can be represented as:


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2min bA −x

x , mxnR∈A , m>n (2.8)These discrete systems are considered ill posed if the singular values of the matrix A decay gradually to zero and the ratio between the highest and the lowest singular values is too high (Hansen, 2007). Ill posed problems in rheology have been widely studied and many methods for the calculation of the relaxation time spectrum had been proposed (Honerkamp and Weese,1989; Baumgaertel and Winter, 1989; Baumgaertel et. al, 1990; De Rosa and Winter, 1994; Winter and Mours,1997; Winter,1997; Roths et al., 2000; Jensen, 2002; Makin and Kuznetsov, 2000; Stadler and Bailly, 2009). Winter et al. (Baumgaertel and Winter, 1989; Baumgaertel et. al, 1990; De Rosa and Winter, 1994; Winter and Mours,1997; Winter,1997) have performed one of the most extensive research to calculate relaxation time spectra for gelling systems in the liquid solid transition. They avoid the resolution of the ill posed problem by calculating the discrete relaxation time spectrum with only a few Maxwell elements and using what they called the parsimonious spectrum (Baumgaertel et. al, 1990). They found that the relaxation time spectrum has local modes, associated to Rouse and De Gennes molecular models. It also displays intermolecular modes, related to crosslinking andliquid solid transition. Thus, they split the spectrum in a short relaxation times zone, named Baumgaertel-Shausberger-Winter (BSW), and a long relaxation times zone, named Winter-Chambon (WC) (Winter and Mours, 1997). Details of the resolution are not specified and commercial software is needed to calculate the spectrum. Honerkamp and Weese (1989) solved ill posed problems by minimization of cost functions using singular values decomposition (SVD) with Tikhonov (Honerkamp and Weese, 1989; Orbey, 1991) or edge preserving regularization (EPR) (Roths et al.,2000; Roths et al. 2001), in order to obtain relaxation time spectra that fit with the BSW and WC curves. A generalized regularization method (Roths, 2001) fits the relaxation time


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spectrum very well if the frequency range is large enough, and the physical interpretation of those spectra are related to the BSW and WC modes, leading to the same relaxation time spectrum shape for all data regardless of the frequency range. A self-consistence routine is performed to determine the regularization parameter. There are other minimization methods to the cost functions, for example Simulated Annealing, which uses Monte Carlo algorithms to obtain an approximation to the global minimum. The Simulated Annealing method leads to a good approximation to the global minimum in a large search space by analogy to thermodynamic annealing processes, where different minimum energetic states can be achieved depending on the cooling rate (Jensen, 2002). This method usually needs long calculation times and is used to find the solution by making random iterations which allow the algorithms to escape from local minima. A simplified method to calculate the relaxation time spectrum was presented by Malkin and Kuznetzov (2000), which linearized the problem in order to use a simple root median square. Nevertheless, this method required a small number of Maxwell elements to avoid the ill posed condition. Recently, Stadler and Bailly (2009) noted that the spectra should be continuous and positive to preserve physical meaning. Thus, they proposed a method based on splines to smooth the spectra obtained by minimum squares. This method needs to establish some adjustable points in the spectrum, named knots, introducing some complexity to the calculation. The number of adjustable points is adjustable, and the spectrum is smoothed using several Hermite splines. The spectrum could be fit to a reasonable physically meaningful time window, and the relaxation time range might be smaller than the inverse of the experimental frequency range. The authors have pointed out that some parts of the continuous spectrum do not influence the fit to the modulus and are thus mere artifacts. In this article, an alternative method based on Thikonov regularization is proposed to solve ill posed problems in order to calculate the relaxation time spectrum by


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simultaneous minimization of a cost function, as the normalized standard deviation, and a restriction function or gradient function of the Karush-Kuhn-Tucker condition. The method allows us to determine the regularization parameter which minimizes the distance between these two minimizations and generates a relaxation time spectrum which highlights the relaxation modes and could be related to the physical changes occurring during the association process in polymeric systems. The method can be applied to isothermally determined data for systems where master curves cannot be generated by time-temperature superposition principle. It will be shown how the frequencies ranges may affect the spectrum and how short frequency range experimental data may be used somehow to interpret changes in complex rheological systems. Proposed Method

As all regularization methods for solving ill posed problems of this kind, the aim of the proposed method is to find a vector (λi,Hi) which satisfies equations (2.3) and (2.4) simultaneously and minimizes equation (2.6) . This can be achieved by linearizing and regularizing the problem in order to obtain an equation that can be minimized. Linearization The first step is to represent the problem linearly as a form of equation (2.8). In order to achieve this, the matrix A from equation (2.7) will be given by Kij, a matrix that contains n Maxwell elements for both G’ and G’’ as follows:


86 December 2015

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

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( ) ( ) ( )

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( ) ( ) ( )

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+

+

+

+

+

+

+

+

+

+

+

+

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11

1...

11

nm

nm

m

m

m

m

n

n

n

n

m

m

m

m

m

m

n

n

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

λωλω

ijK (2.9)

or, 22

22

1 ij

ij

λωλω

+=ijK For i=1,...n, j=1,..m; and

221 ij

ij

λωλω

+=ijK For i=1,...n; j=m+1,...2m

The size of the Kij matrix is 2m x n because it contains the transformation of both G’ and G’’ by the generalized Maxwell’s model. The vector b of equation (2.7) then becomes: mGGb ejj /' −==b For j=1,..m; and b=bj=G’’j For j=m+1,...2m or,


87 December 2015

−

−

−

=

3

2

1

2

1

''

''''

'

'

'

G

GG

mGG

mG

GmGG

em

e

e

b

(2.10)

And finally, vector x (also from equation 2.7), becomes the vector H=Hi(λi). Thus, the equivalent form of equation (2.8) is: 2

.min bHKH

−ij (2.11)Regularization To solve linear ill posed problems, a regularization method is usually employed. Roths et al. (2001) have presented a generalized regularization method to solve ill posed problems. This method consists in the inclusion of a new term Ω(x) in equation (8) or (11) to find a new perturbed solution x’, that is closer to the real solution x with some distinctive characteristics. The new solution can be written as:

2

2

2

2)(min' xxx Ω+−= αbA (2.12)


88 December 2015

Here, α is the regularization parameter which weights the perturbation of the new solution x’. If α were zero, the solution would be the same x with no perturbation, and if α were too high, the perturbed solution x’ would be very different from the real solution. The method proposed in this work is regularized by a second differential, i.e, Ω(x) = Δ2H, in the form of a linear operator L (n x n)= tridiag (1,-2,1) as follows:

−

−−

−

=

120000001210001210001200001

L (2.13)

or, iH2Δ=⋅ HL Using the second derivative as a regularization parameter, the solution is constrained to have a smooth behavior with no abrupt changes, and the relevant equations are:

+−= 2

22.min

λα

dHd i

ij bHKHH (2.14)

2

2

2

2.min L.HbHKH ijH

α+−= (2.15)The regularization operator, as shown by Roths et al. (2000,2001) could also be the edge preserving regularization operator, which is shown in equation (2.16)


89 December 2015

2

2

22

2

2

1

+

=Ω

λϕ

λ dHd

dHd ii

EPR(2.16)

Using the edge preserving regularization operator implies the inclusion of a new optimization, ϕ, parameter that should be found along with the regularization parameter α. The edge preserving regularization method allows to solve spectra with sharp edges or large curvature (Roths et al. 2000). Values of ϕ=0 returns the operator to the Thikonov case. The alternative method proposed in this work could be performed either with Thikonov regularization operator or the EPR. In both cases, the aim is to find the regularization parameter that best represents spectrum. This work will not be focused on the difference between these two methods. Relaxation time range One of the fundamental problems when trying to obtain the relaxation time spectrum is the time range chosen for the calculations. The viscoelastic behavior studied as a generalized Maxwell model would be the result of the contributions of all Maxwell elements together. It is not clear if from dynamic rheometry experiments it would be possible to measure the contribution of those elements with relaxation times too short or too long for the experimental frequencies used. This problem has been extensively studied by Davies and Anderssen (1997,1998), who demonstrated that the relaxation times range is not defined by the inverse of the maximum and minimum experimental frequencies. In their sample and localization theory, they demonstrated that the relaxation time range is defined by the expression:

21min

21max ..

ππ

ωλω−

−− ≤≤ ee i (2.17)


90 December 2015

This implies that dynamic rheology measurements should be performed in a frequency range of several orders of magnitude to obtain a representative relaxation time spectrum. Nevertheless, the expression (2.17) has been obtained from Fourier transforms to solve the inversion problem assuming the relaxation modulus is a function of Gauss error functions, named sampling distributions, and it is always linked to the experimental frequencies chosen. In opposition to the sampling localization theory, Jensen (2002) demonstrated, using simulated annealing to calculate the relaxation time spectrum, that using longer relaxation times ranges it is possible to obtain lower values of the cost functions. Although Jensen obtained better fits with longer relaxation times ranges, the use of a limited relaxation time range is still controversial since it is possible that some information about the relaxation modes could be missing when the experimental frequency range is too short. In this work, the relaxation times range is set using the parameter B, as defined by Jensen (2002) and shown in equation (2.18): max

2/

min ωλ

πBe= ; min

2/

max ωλ

πBe−

= (2.18)where ωmax and ωmin represent the maximum and minimum experimental frequencies. The value of B=0 represents the criteria of the inverse frequencies, and B=1 represents the sampling localization theorem proposed by Davies and Anderssen. There should be an optimal value of B that minimizes the cost function, and also the constrain, as will be shown later. If B is allowed to be lower than zero, e.g., longer relaxation times ranges, it is possible to consider those experimental data with shorter frequency range, i.e., two orders of


91 December 2015

magnitude. Thus, relaxation time spectra can be obtained even for those experimental data sets on which no time-temperature superposition principle can be applied, as will be presented later. Questions arise when longer relaxation time ranges are used, because of their physical interpretation. From our point of view, all relaxations elements in the sample contribute to the final rheological behavior, although it is also clear that it is not possible to get the real information of those contributions beyond the inverse of the frequency range. Nevertheless, if it would be possible to obtain a parameter B<0, which minimizes equation (2.12) and also satisfies the constrain, the spectrum could represent the rheological behavior of the sample inside the experimental range, even if outside this range, the spectrum might be considered as an artifact. By calculating the spectrum using this method for several well-known samples, where physical changes are already characterized, it is possible to validate the method. Minimization For methods based on the regularization to solve the ill posed problem, the solution shall be restricted to the minimization of equations (2.14) or (2.15). In the method proposed here, the solution is constrained using the regularization term as a Lagrange multiplier. Then, the solution is expressed by eq. (2.15) such that,

Ω=H

H min (2.19)The spectrum Hi(λi) obtained by using this constrain is smooth enough to distinguish all the relaxation modes present. To accomplish the minimization of two functions simultaneously, the method uses constrains based on the Karush-Kuhn-Tucker (KKT) conditions (Andreani et al.,2005), i.e., minimizing the first gradient of the function (2.16). This routine can be performed


92 December 2015

using a nonlinear constrained multivariable optimization solver in ATOMS module of Scilab® opensource software. This optimization method is called “ATOMS/fmincon” Let: )()( Hg Ω=H (2.20)

and 2

2

2

2)()( Hf Ω+−= αb.HKH ij (2.21)

with an additional linear constrain to make the solution closer to the real unperturbed solution as follows: bHKH ij −⋅=)(h (2.22)

Then, the KKT condition implies the minimization of f(H) with the constrain: )()()(min)( HHHH hgfG ∇+∇+∇= (2.23)

The solution should then be the vector (H, α, Ge) that minimizes the intersection of the two surfaces formed by the functions f(H) and G(H). If no intersection is found, the solution would be presented as the vector H that minimizes the difference between the two surfaces. The algorithm requires an initial vector H0, to evaluate the function f(H0) in equation (2.21) and the function G(H0). Then, a new value is generated as H0+ΔH, and the algorithm evaluates again the function f(H0+ΔH) and G(H0+ΔH). If the values of f(H0+ΔH) ≤ f(H0) and G(H0+ΔH) ≤ G(H0), the algorithm accepts the vector H0+ΔH as a


93 December 2015

new initial vector for the next iteration. ΔH is iterated until some convergence criterion is achieved. One of the main reasons to use this method is that the Lagrange multiplier, which is in this case the regularization parameter, α, might be evaluated and optimized for each iteration of the vector H, when normally the regularization methods perform a progressive iteration of the regularization parameter to search the vector H, that satisfied the constrains. Regularization parameter

As stated above, the regularization parameter α, weights the perturbation generated by the regularization term introduced in the cost function. To find the optimum regularization parameter, the algorithm presented in this work uses an alternative calculation based on the L curve method (Hansen, 2007). This method consists in plotting bHK −.log ij vs. )(log iHΩ as functions of the regularization parameter α and then analyze its curvature. The optimum parameter αoptim corresponds to the value where the bHK −.log ij vs. )(log iHΩ plot presents its maximum curvature. Figure 2.1a shows schematically how the regularization parameter filters the values of the bHK −.log ij and )(log iHΩ . For low values of α, the logarithm of the norm in the second derivative, )(log iHΩ becomes too large, resulting in a noisy or less filtered spectrum with a very good data fit. On the other hand, large values of α result in a smooth spectrum with large values of the cost function or a bad fit. The point of maximum curvature represents the equilibrium between filtering the spectrum and minimizing the cost function. Based on the L curve method, an alternative method is proposed here, where the norm of the vector formed by b.HKij − and )( iHΩ , normalized from 0 to 1, is calculated for each value of the regularization parameter α. The equilibrium between


94 December 2015

filtering and minimization is given by the regularization parameter having the minimum value of the norm: αα =optimum such that,

norminorm

H )(min 2 Ω+−b.HKijH

(2.24) Figure 2.1b shows the modified L-curve method proposed to find the optimum regularization parameter for the case of Thikonov regularization parameter

HHi2)( Δ=Ω .

Using the modified L-curve method in this way, allows the use of a non-progressive iteration of the regularization parameter similar to that performed by the KKT.

Figure 2.1a). L curve method for progressive iteration of α b) Modified L-curve method proposed in this work for non-progressive iteration of α. Spectrum normalization The method used in this work presents the relaxation time range in a logarithm scale, thus the relaxation time spectrum must be normalized to preserve the contribution of each Maxwell element in the distribution by dividing by ho, where ho is:

bHKij i −.

iH2Δ

α2

α j

α3

bHKij i −.

iH2Δ

α2α 1

α jαoptimum

α i

α3

Less filteredBest fit

SmoothWorse fit

bHKij i −.

iH2Δ

α2

α j

α3

bHKij i −.

iH2Δ

α2α 1α 1

α jαoptimumαoptimum

α iα i

α3

Less filteredBest fit

SmoothWorse fit

bHKij i −.

iH2Δ

α2

α1

α jαoptimum

αi

α3

bHKij i −.

iH2Δ

α2

α1

α jαoptimum

αi

α3

norm

normbHKij i −.

iH2Δ

α2

α1

α jαoptimum

αi

α3

bHKij i −.

iH2Δ

α2

α1

α jαoptimum

αi

α3

norm

norm

a) b)


95 December 2015

1

min

max10 −

= noglho

λλ (2.25)

Application of the method for simulated spectra

The studies of the different methods performed to calculate the relaxation time spectrum from dynamic rheometry are usually tested using a simulated spectrum. One of the most frequently used spectrum is the one presented by Honerkamp and Weese (1989), which is a double Gaussian spectrum defined by equation (2.26): ( ) ( )

−−+

−−=2

)ln()ln(exp

2)ln()ln(exp

.221 22

yixiiH

λλλλπ (2.26)

where λx = 0.05 s corresponds to the first relaxation mode and λy=5 s corresponds to the second maximum. From this simulated spectrum, the data of the dynamic moduli can be generated by means of equations (2.3) and (2.4). This data is used to verify the method proposed here. The spectrum obtained must have the same relaxation modes as the original in equation (2.26).


96 December 2015

Figure 2.2. Storage and loss moduli generated from the simulated spectrum using equations (2.3) and (2.4). Continuous lines indicate the fit with the proposed method for different number of Maxwell elements. Figure 2.2 shows the generated moduli from simulated spectrum and the fits using the proposed method with different amounts of Maxwell elements, n. The goodness of the fit can be quantified by different cost functions. In this case, the standard deviation σ has been used and is given by equation (2.27): ( ) ( ) ( )( )bHK ij −⋅

=−+−

=

=

)(.1''''''.1 2

1

2 ασm

GGGGm calcidatai

m

icalcidatai (2.27)

where, m is the number of data points or experimental frequencies, G’idata and G’’idata are the experimental data (in this case the generated data) of the moduli for each frequency and G’calc and G’’calc are the moduli calculated with the proposed method. Figure 2.3 shows the simulated spectrum from equation (2.26) and several spectra calculated using the method proposed in this work with different number of Maxwell

1E-3 0,01 0,1 1 10 100 10001E-5

1E-4

1E-3

0,01

0,1

1

G

',G''

Frequency (ω)

G' generated G'' generated G' n=30 G' n=60 G' n=90


97 December 2015

elements n. It can be seen that all spectra have the same two relaxation modes. Smaller number of Maxwell elements leads to a less representative spectrum.

Figure 2.3. Simulated spectrum from Honerkamp and Weese (1989) and the calculated from the generated data using the method proposed in this work. Table 1. Goodness of the fit using the method proposed in this work for different number of Maxwell elements. Number of Maxwell elements Standard deviation σ Regularization parameter α 18 7.86E-03 1.00E+0024 2.13E-03 4.42E-01 30 3.49E-04 2.44E-01 36 2.88E-04 9.36E-02 42 1.56E-04 5.43E-02 48 9.43E-04 3.13E-02 54 8.65E-04 5.88E-03 60 3.36E-04 5.65E-03 90 1.33E-04 4.22E-04 120 1.14E-04 4.84E-05

1E-3 0.01 0.1 1 10 100 1000

0.00

0.05

0.10

0.15

0.20

Hi/h

o

Relaxation time λ

simulated n=18 n=24 n=30 n=60 n=90


98 December 2015

Table 1 shows the values of the cost function for the fit performed to the generated data with the proposed method and different amounts of Maxwell elements n and the regularization parameter α calculated for each fit. It can be seen that for larger number of Maxwell elements n, the values of the cost functions are smaller, as well as the regularization parameter α. However for more than 42 Maxwell elements the decrease in standard deviation is not significant. The regularization parameter was calculated with the modified L curve method as shown in Figure 2.4 for 60 Maxwell elements, and the minimum vector shows the optimum value of the regularization parameter. Each point represents one value of the regularization parameter.

Figure 2.4. Modified L curve to determine the optimum regularization parameter Figure 2.5 shows how the standard deviation varies with the number of Maxwell elements introduced for the calculation of the relaxation time spectra. There exists a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

||Δ2 H

|| norm

||KijH-b||norm

αoptimum=0.0057


99 December 2015

clear breakpoint in the number of Maxwell elements from which the value of the standard deviation is stabilized. This point may be taken as an optimum number of Maxwell elements. This behavior has similarity with the parsimonia spectrum proposed by Winter (1997). However, he keeps the number of Maxwell elements low enough to avoid the ill posed problem. In this case, the number of Maxwell elements is always above the ill condition and are optimized with the standard deviation. Both regularization operators have been used for comparison purposes, and it can be seen that both methods have the same inflection point in the standard deviation plot of Figure 2.5.

Figure 2.5. Standard deviation as a function of the number of Maxwell elements for the two regularization operator Smoothing and interpolation Another problem associated with the calculation of relaxation time spectra is related with the data noise or experimental errors. In general, experimental errors may be determined statistically from several assays and the error associated with every

0 20 40 60 80 100 120

0,000

0,005

0,010

0,015

0,020

Ω EPR Ω Thikonov

Stan

dard

dev

iatio

n σ

number of Maxwell elements


100 December 2015

frequency might be calculated. The experimental data should be averaged in order to consider all errors. In this work, the experimental data could be smoothed using either a moving average filter function or a Savitzky-Golay filter function. These functions generate an average series of data subsets to produce a smooth curve. The algorithm for the spectrum calculation considers this smooth curve as the new experimental data. Data could be also interpolated without affecting the final spectrum. To verify the influence of interpolation, different data were generated from the Honerkamp spectrum at different number of frequencies, m. In all cases, the interpolation was performed to obtain 60 points from the data, before calculating the relaxation time spectrum. Figure 2.6 shows the resulting spectra for these interpolations, where basically no influence in the spectrum was observed above m=20, which means 3,3 points each decade. Thus, if the experimental data is above 4 points per decade, the interpolation might be useful. Nevertheless, even knowing that some points of the experimental data have no effect over the spectrum, it is not possible to take it for granted, then it would be important to have as many data points as it is experimentally possible, in order to get a more detailed relaxation time spectrum.

Figure 2.6. Effect of the number m of data points in the spectrum calculation 1E-3 0,01 0,1 1 10 100 1000

0,00

0,05

0,10

0,15

0,20

0,25

Hi/h

o

Relaxation time λ

simulated m=15 m=20 m=30 m=60


101 December 2015

Pseudo-code

Constrained optimization is accomplished by using an optimization tool “ATOMS/fmincon” available in Scilab®. Several optimizations are performed with this tool where the outputs arguments are the vector Hi, Ge value (if required) and the optimization parameter lambda α. Input arguments are the matrix Kij and the vector of b=(G’,G’’). The algorithm allows to smooth and interpolate the input data, and also to choose the regularization operator (Thikonov or EPR). The initial vector H0 could be obtained from the general singular value decomposition method (GSVD) (Honerkamp and Weese 1989) and the α0 by using the L curve method (Hansen, 2007). begin Input data Choose method \\ Thikonov or EPR While Standard deviation σ> termination criteria do for n0= 2:end do \\ number of Maxwell elements per decade for B=0: -3 do\\ expansion of the relaxation time range initialization Generate H0 α0 \\ generated by GSVD method and L curve for i:end do\\ number of iteration of regularization parameter α Calculate )()( ii Hg Ω=H Calculate 2

2

2

2)()( Hf iii Ω+−= αb.HKH ij Calculate h(H)=Kij.H-b Used fmincon to calculate Hi and αi to Minimize

f(H) and h(H) with constrained )()()(min)( HHHH hgfG ∇+∇+∇= Used αi +1=αi to calculate next Hi+1

Saved Hi,αi until i=end. Calculate log )( iHΩ and bHK −.log ij Calculate αoptim=norm

inormH )(min 2 Ω+−b.HKijH

end for Calculate standard deviation σ If σ<termination criteria stop


102 December 2015

else next B end if end for If σ<termination criteria stop else next n0 end if end for end While end Experimental frequency range

Using the generated data, it is possible to study the influence of the experimental frequency range in the relaxation time spectrum. Generated moduli data from the simulated spectrum were truncated to obtain three different frequency ranges that are shorter than the original. The spectrum resulting from the application of our proposed method to the truncated data should not differ from the simulated through equation (2.26). For this propose, the first truncated data set was selected from the generated moduli data going from 0.001 rad/s to 0.7882 rad/s and will be referred here as low frequency data. The second truncated data set was selected in a frequency range from 0.1172 rad/s to 92.37 rad/s, this data set will be referred here as middle frequency data. The third truncated data set, referred as high frequency data, was taken from 1.269 rad/s to 1000 rad/s. Figure 2.7 (a to c) shows the fits for the three different truncated data sets. From these results it can be concluded that our method can be applied to experimental data with shorter frequency ranges and the main relaxation modes may still be predicted.


103 December 2015

Figure 2.7. Truncated data generated from the simulated spectrum using Equations (3) and (4). Filled symbols are G’ and open symbols are G’’. a) Low frequency data set, σ= 3.73 x 10-4, α=3.73 x 10-4; b) Middle frequency data set, σ=5.99 x 10-4, α=1.97 x 10-3 c) High frequency data set, σ=2.64 x 10-4, α=7.37 x 10-4 d) Short frequency range,

σ=7.19 x 10-4, α=5.84 x 10-4 For all the truncated data, the full relaxation times spectra were calculated and they are shown in Figure 2.8 (a to c). The middle frequency truncated data set generates a spectrum which is almost the same as the original. For the low frequency data set, the first relaxation mode seems to be largely decreased or missing. This means that some information regarding this relaxation mode has not been taken from the input data. Nevertheless, the second mode appears at the same point as in the simulated data. For the high frequency data set, the first relaxation mode appears at the same position, but the longest relaxation mode has decreased in importance (as judged by its relative intensity), however it appears at the same relaxation time as the simulated one. This suggests that high frequency data contribute more to build the spectrum than the low frequency data. This conclusion may be related with the presence of inflexions points in the data.

1E-3 0.01 0.1 11E-5

1E-4

1E-3

0.01

0.1

1

G',

G''

(Pa)

Frequency (rad/s)0.1 1 10 100

0.01

0.1

1

G',

G''

(Pa)

Frequency (rad/s)

1 10 100 10001E-3

0.01

0.1

1

G',

G''

(Pa)

Frequency (rad/s)0.1 1 10 100

0.01

0.1

1

G',

G''

(Pa)

Frequency (rad/s)

a) b)

c) d)


104 December 2015

The data were also truncated as shown in Figure 2.7d to a convenient shorter range to test if the method can still reproduce the simulated spectrum. This new data were taken in a frequency range encompassing 0.489 rad/s to 57.4 rad/s, which is a typical experimental frequency range employed to determine rheological properties of polymer solutions. Figure 2.8d shows the spectrum generated from these data. The presence of the two relaxation modes is consistent with the simulated spectrum from Honerkamp and Weese. Even if the frequency range is shorter, the spectrum looks more like the simulated one than those obtained from low frequency or high frequency truncated data sets. Thus, a conclusion derived from these results is that the relaxation modes or maxima in the relaxation time spectrum are related with inflexions points in the frequency sweep. It is important to point out that all spectra from truncated data sets were calculated using a relaxation time larger than the inverse of the minimum and maximum frequencies of the experimental data. This point will be discussed in the next section.


105 December 2015

Figure 2.8. Relaxation time spectrum for moduli data from the simulated spectrum (solid black lines), and from similar simulated spectra but truncated at three different intervals (solid grey lines). a) Low frequency data set; b) Medium frequency data set; c) High frequency data set; d) Short frequency data set. Arrows represents the inverse of the experimental frequency data Relaxation time range

The sampling localization theorem of Davies and Anderssen (1997,1998) defines the relaxation time range as a function of the maximum and minimum experimental frequencies, as it was specified in equation (2.17). This theorem shows that there is a single relaxation time range and it is shorter than the inverse of the maximum and minimum experimental frequencies. On the other hand, Jensen (2002) showed that using ranges larger than the inverse of the experimental frequencies leads to better fits. Equation (2.16) taken from Jensen work, can be used to test Jensen hypothesis.

d)

a) b)

c)

1E-3 0.01 0.1 1 10 100 1000

0.05

0.10

0.15

0.20

Hi/h

o

λ (s)1E-3 0.01 0.1 1 10 100 1000

0.00

0.05

0.10

0.15

0.20

λ (s)

Hi/h

o

1E-3 0.01 0.1 1 10 100 1000

0.00

0.05

0.10

0.15

0.20

λ (s)

Hi/h

o

1E-3 0.01 0.1 1 10 100 1000

0.00

0.05

0.10

0.15

0.20

Hi/h

o

λ(s)


106 December 2015

Figure 2.9 shows the spectra calculated for different values of B, i.e., different ranges of the relaxation time for the short frequency range data of Figure 2.7d. It can be seen that expansion of the relaxation time spectrum range benefits the fit and the calculation. This is an important finding, because knowing the real spectrum is difficult to argue that this expansion beyond the inverse of frequencies could be considered as simple artifacts. All of the spectra show two relaxation modes that are similar to those in the original simulated spectrum and there is no significant difference between spectra for B<-1. The number of Maxwell elements per decade has been fixed to 10, in order to normalize all spectra.

Figure 2.9.Relaxation time spectra calculated for different values of B normalized with 10 Maxwell elements per decade. a) With B=0; b) B=-1 ; c) B=-2; d) original relaxation time range from 0.001 s to 1000 s Expansion of the relaxation time range must also be linked to the physical phenomena involved.

1E-3 0.01 0.1 1 10 100 10000.00

0.05

0.10

0.15

0.20

Hi/h

o

relaxation time (λ)1E-3 0.01 0.1 1 10 100 1000

0.00

0.05

0.10

0.15

0.20

relaxation time (λ)

Hi/h

o

1E-3 0.01 0.1 1 10 100 1000

0.00

0.05

0.10

0.15

0.20


Hi/h

o

1E-3 0.01 0.1 1 10 100 1000

0.00

0.05

0.10

0.15

0.20


Hi/h

o

a)

c)

b)

d)


107 December 2015

Winter and Mours (1997) quoted that the spectrum obtained from rheological measurements should not be valid at short relaxation times, where the glass transition and simple entanglements become important. Polymer chains, regardless of their macroscopical complexities (as polymer gels), should have a glass transition. On the other hand, higher relaxation times might be related with intermolecular associations which retard the movement of polymer chains, showing higher relaxation modes. Crosslinking structures above critical gel point for example are not allowed to relax completely and the relaxation time spectrum should divert. There are other viscoelastic systems that may have complexities in the relaxation of their structures. That would be the case of surfactant/polymer systems, which present synergistic effects on their viscoelastic behavior (Rodriguez et al. 2011, Shashkina et al. 2005, Rojas et al 2010, Gouveia et al. 2009, Panmai et al. 1999, Peiffer 1999, Couillet et al. 2005). In such cases, different relaxation modes should be present in their spectra, because of the different types of associations present in the structures. The physical phenomena involved in the experiment must be considered to choose the optimum relaxation time range. Relaxation time expansion, as suggested in equation 18, is symmetrical with respect to the original inverse of the minimum and maximum frequencies. Non symmetrical approaches could also be calculated, as in the case of Figure 2.10d, where the relaxation time limits are the same of the original data. Relaxation spectra from experimental data

Until this point, our method has been validated using a simulated spectrum, nevertheless for experimental cases the real spectrum only exists as a mechanical model and it cannot be assumed as known. In the literature some rheologically complex systems whose behavior have been sufficiently characterized can be found. Wormlike micelles systems are normally


108 December 2015

rheologically active above some critical concentration, where interaction between micelles increases the viscosity of the system giving the solution a viscoelastic behavior (Rodriguez et al. 2011, Shashkina et al. 2005, Rojas et al 2010). Moreover, it has been extensively studied that these interactions between wormlike micelles might be reinforced by adding a polymer that might increase those interactions (Rodriguez et al. 2011, Shashkina et al. 2005, Rojas et al 2010). Hydrophobically modified, anionic or zwiterionic polymers could be added in small quantities to increase the viscoelastic behavior of a cataionic rodlike micellar solutions like cetyltrimetylammonium p-toluene sulfonate (CTAT) in water (Rodriguez et al. 2011). CTAT has a critical micellar concentration (CMC) of 0.26 mM and a critical rodlike concentration CRC of 1.97 mM (Rodriguez et al. 2011). Above a threshold concentration (11mM), in the semi-dilute range, the micelles start to entangle under static conditions because of the growth of rodlike micelles to become large wormlike micelles. Above this concentration, the surfactant solution behaves as a viscoelastic fluid because of the micellar interactions (Rodriguez et al. 2011). The experimental data used in this work to calculate relaxation time spectra are from 20mM CTAT solutions to which different concentrations of a zwitterionic polyacrylamide copolymer were added to enhance the viscoelastic behavior of the samples (Rodriguez et al 2011). The increases in viscosity achieved by adding small quantities of this copolymer to a CTAT solution at 20mM are a consequence of specific interactions between the sulfobetaine units and the CTAT wormlike micelles. The copolymer chains aggregates and attaches to the micelles to form physical binding between different micelles (Rodriguez et al). Such complex systems would have different attached points that might relax at different times. The relaxation time spectra expected for this kind of systems should show different relaxation modes, some of them would correspond to simple local entanglements and some others will appear after the addition of the copolymer to the solution, representing the more complex binding structures that join micelles together. Figure 2.10 shows the dynamic rheological data taken from Rodriguez et al.


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(2011). As it can be seen, the experimental frequency range was limited to less than two orders of magnitude. No time-temperature superposition could be performed since critical concentrations (CMC, CRC, etc) change or depend on temperature. Five different concentrations of a zwitterionic copolymer were added to the 20mM CTAT solution to see the changes in viscoelastic behavior. Relaxation time spectra were calculated in this work using the Thikonov regularization parameter and the KKT condition. In these experiments, it was possible to observe the crossover point of the storage and loss moduli within the experimental frequency range. The inverse of the crossover frequency is usually associated with the longest mean relaxation time for the formation or breakup of the microstructure (Rodriguez et al. 2011, Shashkina et al. 2005) At high surfactant concentration, without polymer, the viscoelastic properties of the solution depend on the length of the micelles which could also exhibit reversible breakage while reptating from their constraining tube of entanglements (Shashkina et al. 2005). For such systems, it have been found that the stress relaxation fits the Maxwell model and this behavior is attributed to the fact that the micelle breaking time is short compared to the reptation time (Shashkina et al. 2005). Those systems have a gel like behavior. Adding polymer to the systems increased the viscosity of the sample changing also the viscoelastic properties of the micellar structure. This is due to the formation of a network of surfactants with the zwitterionic copolymer. This structure generates new relaxations modes at higher times, maintaining the same primary relaxation modes of the simple micelle entanglements. As the copolymer concentration rises, more complex structures and bonding appear with different relaxation modes. Figure 2.11 shows the change in the relaxation modes with no polymer added and for five different concentrations of polymers.


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Figure 2.10. Small angle dynamic rheometry for a solution of 20mM of CTAT and different concentration of a zwitterionic copolymer (PAM-Z). Taken from Rodriguez et al (2011). Lines represent the fit after calculating the relaxation times spectra with the method proposed.

1 10 1000.1

1

G',

G''

(Pa)

Frequency (1/s)1 10 100

0.1

1

G',

G''

(Pa)

Frequency (1/s)

1 10 1000,1

1

G',

G''

(Pa)

Frequency (1/s)1 10 100

0,1

1

G',

G''

(Pa)

Frequency (1/s)

0,1 1 10 1000,1

1

G',

G''

(Pa)

Frequency (1/s)0,01 0,1 1 10 100

0,1

1

G',

G''

(Pa)

Frequency (1/s)

0 ppm

800 ppm

500 ppm

1000 ppm

1500 ppm 2000 ppm


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Extension of the relaxation time range was performed for all the calculated spectra. The coincidence in the first relaxation mode for all samples is remarkable. A broadening of the distribution of relaxation times could be seen when polymer is added from 500 and 800 ppm. This is a consequence of formation of heterogeneous assemblies between CTAT micelles. Increasing the polymer concentration to 1000, 1500 and 2000 ppm cause the formation of new relaxation modes at longer times. A broad higher relaxation mode appears approximately at 20 seconds when 2000 ppm of zwitterionic copolymer is added to the CTAT solution. Those large scale modes should be interpreted as new intermicellar association from the synergistic interactions between polymer and surfactant, which bind the worm like micelles together and reduce their capability to relax. Thus, calculating the relaxation time spectrum allows the relaxation modes of complex assemblies in the structure to be differentiated.

Figure 2.11. Relaxation time spectra calculated from data of a 20mM CTAT solution with different concentration of a zwitterionic copolymer Conclusions

It is possible to obtain the relaxation time spectrum from the storage and loss moduli determined by small angle dynamic rheometry in the linear viscoelastic range, by simultaneous minimization of a cost function, like the regularized standard deviation,

1E-3 0.01 0.1 1 10 100

0

1

2

3

4

Hi/h

o

Relaxation Time (λ)

0ppm 500 ppm 800 ppm

local modesingle entanglements

1E-4 1E-3 0.01 0.1 1 10 100 1000 10000

0

1

2

3

large scale synergistic entanglements

Hi/h

o

Relaxation Time (λ)

1000 ppm 1500 ppm 2000 ppm

local modesingle entanglements


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and a restriction function or gradient function of the Karush-Kuhn-Tucker condition. The method allows us to perform the regularization either with a Thikonov regularization operator or with the EPR. The aim is finding the right regularization parameter that satisfies a good data fit and a smooth relaxation time spectrum. The spectrum obtained should be smooth and continuous. Despite the controversial used of low frequency range data to obtain relaxation time spectrum, it has been shown that expanding the range of relaxation time spectrum might enhanced the accuracy of the calculation. The expansion of the relaxation time range allows the calculation of relaxation spectra from experimental data that spans a short frequency range, in those cases where no time-temperature superposition can be applied. For complex associated viscoelastic fluids, it would be possible to distinguish between the different structures present in the system, as long as they exhibit different relaxation modes. References

ANDREANI, R.; MARTINEZ J.M, and SCHUVERDT M.L. On the relation between constant positive linear dependence condition and quasinormality constrain qualification. Journal of Optimization Theory and Application 2005, vol. 125, p.p 473-485. BAUMGAERTEL, M.; and WINTER, HH. Determination of Discrete Relaxation and Retardation Time Spectra from Dynamic Mechanical Data. Rheologica Acta, 1989, vol. 28, no. 6, pp. 511-519. BAUMGAERTEL, M.; SCHAUSBERGER, A.and WINTER, HH. The Relaxation of Polymers with Linear Flexible Chains of Uniform Length. Rheologica Acta, 1990, vol. 29, no. 5, pp. 400-408. COUILLET I., et al.Synergistic effects in aqueous solutions of mixed wormlike micelles and hydrophobically modified polymers. Macromolecules 2005;vol. 38,pp. 5271-5282


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DAVIES, AR; and ANDERSSEN, Robert Scott. Sampling Localization in Determining the Relaxation Spectrum. Journal of Non-Newtonian Fluid Mechanics, 1997, vol. 73, no. 1, pp. 163-179. DAVIES, AR; and ANDERSSEN, RS. Sampling Localization and Duality Algorithms in Practice. Journal of Non-Newtonian Fluid Mechanics, 1998, vol. 79, no. 2, pp. 235-253. DE ROSA, ME; and WINTER, HH. The Effect of Entanglements on the Rheological Behavior of Polybutadiene Critical Gels. Rheologica Acta, 1994, vol. 33, no. 3, pp. 220-237. FERRY, John D. Viscoelastic Properties of Polymers. Third ed. New York: John Wiley & Sons, Inc, 1980. GOUVEIA L.M.;, GRASSL B. and MÜLLER A.J. Synthesis and rheological properties of hydrophobically modified polyacrylamides with lateral chains of poly(propylene oxide) oligomers. Journal of Colloid and Interface Science, 2009; vol. 342, pp.103-109 HANSEN, Christian. Regularization Tools Version 4.0 for Matlab 7.3. Numerical Algorithms, 2007, vol. 46, no. 2, pp. 189-194. HONERKAMP, J.; and WEESE, Jürgen. Determination of the Relaxation Spectrum by a Regularization Method. Macromolecules, 1989, vol. 22, no. 11, pp. 4372-4377. JENSEN, Erik Appel. Determination of Discrete Relaxation Spectra using Simulated Annealing. Journal of Non-Newtonian Fluid Mechanics, 2002, vol. 107, no. 1, pp. 1-11. MALKIN, Alexander Ya; and KUZNETSOV, VV. Linearization as a Method for Determining Parameters of Relaxation Spectra. Rheologica Acta, 2000, vol. 39, no. 4, pp. 379-383. ORBEY, Nese; and DEALY, John M. Determination of the Relaxation Spectrum from Oscillatory Shear Data. Journal of Rheology (1978-Present), 1991, vol. 35, no. 6, pp. 1035-1049. PANMAI S. ; PRUD’HOMME D.G. andPEIFFER D.G. Rheology of hydrophobically modified polymers with spherical and rod-like surfactant micelles. Colloids Surface A: Physicochemical and Engineering Aspects, 1999; vol. 147, pp. 3-15 PEIFFER D.G. Hydrophobically associating polymers and their interactions with rod-like micelles. Polymer, 1990, vol. 31, pp.2353-2360 RODRÍGUEZ M., et al. Shear rheology of anionic and zwitterionic modified polyacrylamides. Colloids and Surface A: Physicochemical and Engineering Aspects, 2011, vol. 373, pp. 66-73


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ROJAS M.R.; MÜLLER A.J. and SÁEZ A.E. Synergistic effects in flows of mixtures of wormlike micelles and hydroxyethyl celluloses with or without hydrophobic modifications. Journal of Colloid and Interface Science, 2008, vol. 322, pp.65-72 ROTHS, T., et al. A Generalized Regularization Method for Nonlinear Ill-Posed Problems Enhanced for Nonlinear Regularization Terms. Computer Physics Communications, 2001, vol. 139, no. 3, pp. 279-296. ROTHS, T., et al. Determination of the Relaxation Time Spectrum from Dynamic Moduli using an Edge Preserving Regularization Method. Rheologica Acta, 2000, vol. 39, no. 2, pp. 163-173. SHASHKINA J.A., et al. Rheology of Viscoelastic Solutions of Cationic Surfactant. Effect of Added Associating Polymer. Langmuir 2005, vol. 21, pp.1524-1530 STADLER, Florian J.; KASCHTA, Joachimand MÜNSTEDT, Helmut. Thermorheological Behavior of various Long-Chain Branched Polyethylenes. Macromolecules, 2008, vol. 41, no. 4, pp. 1328-1333. STADLER, Florian J.; and BAILLY, Christian. A New Method for the Calculation of Continuous Relaxation Spectra from Dynamic-Mechanical Data. Rheological Acta, 2009, vol. 48, pp. 33-49. WINTER, Henning H. Analysis of Dynamic Mechanical Data: Inversion into a Relaxation Time Spectrum and Consistency Check. Journal of Non-Newtonian Fluid Mechanics, 1997, vol. 68, pp. 225-239. WINTER, H. H.; and MOURS, M. Neutron spin echo spectroscopy viscoelasticity rheologySpringer, 1997. Rheology of Polymers Near Liquid-Solid Transitions, pp. 165-234. WOOD-ADAMS P. and COSTEUX S. Thermorheological Behavior of Polyethylene: Effects of Microstructure and Long Chain Branching. Macromolecules, 2001, vol. 34, pp. 6281-6290


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List of Symbols

A generic matrix α Regularization parameter αoptimum Optimum Regularization parameter B optimization parameter for definition of relaxation times range

b vector with experimental moduli values f(H) Regularized constraint

g(H) Euclidean norm of regularization term G Relaxation modulus G’ Storage modulus G’’ Loss modulus Ge Equilibrius modulus. Gexp Experimental modulus G’exp Experimental storage modulus G’’exp Experimental loss modulus G(H) KKT condition h(H) Additional linear constraint Hi Discrete relaxation time spectrum H Relaxation time spectrum vector ϕ Relaxation time Kij Kernel matrix L second derivative matrix operator λi Relaxation time


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λmax Maximum relaxation time λmin Minimum relaxation time m number of experimental frequencies n number of relaxation times γ Strain t time τ stress ω Frequency ωmin Minimum experimental frequency ωmax Maximum experimental frequency Ω Regularization operator x generic solution vector

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CHAPTER III

Dynamic Rheology and Relaxation Time

Spectra of Oil-Based Self-degradable

Gels

Oscar Vernáez, Sylvie Dagreou, Bruno Grassl, A.J Müller

Doctoral student contribution: In this chapter, the student has performed all the experimental section, along with the analysis using the algorithm developed in the previous chapter. He also wrote the paper which was submitted and accepted in the Journal of Polymer Science: Polymer Physics Ed. He is also the corresponding author.

Chapter III: Dynamic Rheology and Relaxation Time Spectra of Oil-Based Self-degradable Gels

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ABSTRACT

In oil well treatments, such as matrix stimulations or water shut-off, it is often necessary to temporary isolate or protect productive zones with chemical diverting agents. In this work, a solution of peroxide crosslinked styrene-butadiene rubber (SBR) has been transformed to a self-degradable gel system by adding hydroperoxide as a degradation agent to the formulation. This oil-based self-degradable gel has been characterized by linear oscillatory rheometry. In-situ and ex-situ experiments were performed to evaluate the evolution of crosslinking and degradation reactions, including the liquid-solid transition. Relaxation time spectra were calculated from dynamic mechanical frequency sweeps. Structural changes in the polymer network were visible within the relaxation time spectra, since it qualitatively showed the contribution of local simple entanglements and chemical covalent bonds to the final rheological behavior. The influence of peroxide concentration, polymer concentration, hydroperoxide concentration and temperature have been studied and described in terms of rheological changes. Finally, a hydrogen donor aromatic solvent was used as scavenger to retard both crosslinking and degradation reactions.


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RESUMEN

En algunas operaciones realizadas en pozos petroleros, como estimulaciones matriciales o control de agua con química sellante, a veces es necesario aislar o proteger temporalmente intervalos productivos con agentes de divergencia química. En este trabajo, una solución de caucho estireno butadieno (SBR) se ha convertido en un sistema gelificante autodegradable con la adición de hidroperóxidos como agente de degradación para la formulación. Este gel base aceite autodegradable se caracterizó mediante reometría oscilatoria. Se realizaron experimentos in-situ y ex-situ para evaluar la evaluación de las reacciones de entrecruzamiento y degradación incluyendo la transición líquido-sólido. Los espectros de tiempos de relajación fueron calculados a partir de barridos de frecuencia en ensayos dinámicos oscilatorios. Los cambios estructurales en la red polimérica se evidenciaron en los espectros de tiempos de relajación, por lo que fue posible mostrar cualitativamente la contribución de los enredos locales simples y los enlaces químicos covalentes al comportamiento reológico. El efecto de la concentración de peróxido, la concentración de polímero, la concentración de hidroperóxido y la temperatura se estudiaron y se describieron en términos de los cambios reológicos. Finalmente, un solvente aromático donador de hidrógenos fue utilizado como un retardante para las reacciones de entrecruzamiento y degradación.


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RÉSUMÉ

Dans les opérations de traitement de puits de pétrole, comme la stimulation au sein de roches, ou la prévention des venues d’eau, il est souvent nécessaire d'isoler ou protéger temporairement des zones de production avec agents chimiques de dérivation. Dans cette thèse, une solution de caoutchouc de styrène-butadiène (SBR) réticulé avec du peroxyde organique, a été transformée en un gel autodégradable par l’ajout d’hydroperoxyde comme agent de dégradation. Ce gel auto-dégradable à base d’huile a été caractérisé en rhéologie linéaire oscillatoire. Des expériences in-situ et ex-situ ont été réalisées afin d’évaluer l’évolution de les réactions de réticulation et dégradation, y compris la transition liquide-solide. Les spectres de temps de relaxation ont été calculés à partir de données de spectroscopie mécanique. Les changements de structure du réseau polymérique ont été mis en évidence par les spectres de temps de relaxation, et nous avons pu montrer qualitativement la contribution d’enchevêtrements locaux simples et de liaisons chimiques covalentes au comportement rhéologique. L’influence de la concentration en peroxyde, en polymère, en hydroperoxyde, ainsi que de la température, ont été étudiées et décrites en termes de changements du comportement rhéologique. Enfin, un solvant aromatique donneur d’hydrogène a été utilisé comme retardateur des réactions de réticulation et de dégradation.

Chapter III : Dynamic Rheology and Relaxation Time Spectra of Oil-Based Self-degradable Gels

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Introduction

The use of organic peroxide as initiators for crosslinking reactions of polydienes is well known. First, radicals formed during initiator decomposition are transferred to polymer chains by hydrogen abstraction. These radicals on the polymer chains can be stabilized by double bond conjugation. For a polymer solution above its critical overlap concentration, the termination of those radicals by combination with each other leads to covalent links, increasing the molecular weight until all polymer chains belong to the same polymer network. Reaction kinetics determines gelation time and the evolution of the mechanical properties of the system. Akiba and Hashim (1997) reviewed the proposed mechanisms for peroxide crossliking of elastomers. Dluzneski (2001) proposed several mechanisms for polydienes crosslinking using organic peroxides as initiators. In anaerobic environments, the principal mechanisms are: homolytic decomposition of peroxide, hydrogen abstraction from polymer chains or any other donor, radical addition to double bonds, termination by radicals coupling or crosslinking, polymer chain scissions and radicals transfer, among others. A more extensive research about the complexity of crosslinking reaction kinetics of elastomers with peroxide, has been performed by Likozar and Kranjc.(2009, 2011) For dilute polymer solutions, it has been demonstrated that the use of hydroperoxides as degradation agents leads to β-scission degradation reactions (Vernáez et al. 2014). It can be assumed that hydroxyl radicals abstract specific hydrogen atoms from the polymer chains, which are susceptible to scission reactions. During chemical treatments in oil wells, such as reactive and non-reactive stimulations or water shut-off, it is sometimes desirable to isolate productive zones to direct the chemical treatment to another particular zone, without affecting the productivity of the isolated region. In such cases, a gel can be placed in the productive zone prior to the chemical treatment. This gel should divert further treatments to the desired zones and must have particular rheological properties to resist bottom-hole pressures during further injections. After the chemical treatment, it is necessary to return the isolated


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zone to its original status. Some commercial gels require further injection of degradation agents to remove the polymer network from the formation matrix, which represents a major cost, and could lead to non-effective removal and deferred production. Another possibility is to design a self-degradable gel system by adding a degradation agent in the initial formulation, which activates after some period of time by temperature increases (Ventresca, et al.). The extension of crosslinking and degradation reactions can be evaluated through the physical changes in the system. In this work, a SBR polymer network is transformed into a self-degradable gel system by incorporating two different radical initiators. The first one has a short decomposition time and acts as a radical initiator for the crosslinking reaction. The second one, with a longer half-life time, decomposes itself slower and acts as a degradation agent by promoting β-scission reactions in the polymer network leading to low molecular weight soluble chains. The competition between crosslinking and degradation reactions results in a self-degradable gel, as shown schematically in Figure 3.1. By changing the concentration of both peroxide as crosslinking agent, and hydroperoxide as degradation agent, it is possible to control the rheological behavior of the gel, depending on temperature, polymer concentration and, as it will be pointed out later, on the concentration of an alkyl substituted aromatic additive in the solvent.

Figure 3. 1. Scheme indicating the competitive reactions of the final self-degradable gel behavior. The green curve refers to the crosslinking reaction, the red curve to the degradation reaction and the black curve refers to the final behavior of simultaneous crosslinking and degradation reactions


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During liquid-solid transition, structural changes occur in the system affecting molecular mobility, leading to large changes in rheological behavior. In classic rubber elasticity theory, the shear modulus is proportional to crosslink density and temperature (Young, 1991), and cure evolution of elastomers can be characterized by rheological changes. Small amplitude oscillatory shear is the method commonly used for systems with a broad distribution of relaxation modes or for materials which undergo changes during the measurement. The time needed for taking a single datum point is in the order of the period of a strain oscillation, which allows measuring a behavior from both longer and shorter relaxation modes, with time constants relatively close to the inverse of the frequency. This means that rheological behavior studied by small amplitude oscillatory shear experiments takes into account the contribution of structural conformation of the network at different scales, even if only a small fraction of the spectrum is actually sampled (Winter and Mours 1997). Conventionally, the gel point is determined by the crossover point of G’ and G’’ in frequency sweep experiments, however this may lead to different gelation times depending on the frequency used (Wang et al., 1997). For better understanding the structural changes occurring during crosslinking reactions, the relaxation modes of the system can be studied by determining the relaxation time spectrum(Wang et al., 1997). Winter et al.(Winter and Mours, 1997; Baumgaertel and Winter, 1989; Baumgaertel et al. 1990) have performed extended research in relaxation time spectra for gelling systems. They found that the relaxation time spectrum has local modes, associated to Rouse and De Gennes molecular models, and also shows intermolecular modes, related to crosslinking and the liquid-solid transition. Thus, they split the spectrum in a short relaxation times zone, named BSW (Baumgaertel-Shausberger-Winter), and a long relaxation times zone, named WC (Winter-Chambon)(Baumgaertel et al. 1990). Curro and Pincus (1983) gave a molecular interpretation of the long time relaxation for elastomers related to De Gennes reptation theory with topological constrains. They based their theory in the distribution of dangling chain ends which leads to a relaxation modulus that obeys a power law dependence on time. From this theory, dangling chain ends lead to extremely long viscoelastic processes.


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The relaxation time spectrum cannot be obtained directly from the oscillatory shear experiments and its calculation implies solving an ill posed problem, e.g., it has several solutions or there are many different arrays of Hi leading to the same relaxation behavior(Vernáez and Müller, 2014). Several methods have been developed to calculate the relaxation time spectrum (Baumgaertel et al. 1990; Vernáez and Müller, 2014; Ferry, 1980; Elster et al. 1991; Brabec and Schausber 1995; Roths et al. 2000; Jensen, 2002; Hansen, 2008; Stadler and Bailly, 2009). Nevertheless, in this work the method developed by Vernáez and Müller (2014)is applied, because experimental data are available in a short frequency range and master curves can be obtained by time-temperature principle, specially for in-situ experiments. This method is based on regularization and the aim is to find a vector (λi,Hi), which satisfies discrete Maxwell equations for storage and loss modulus simultaneously, and that minimizes a cost function as the normalized standard deviation between experimental data and the model. In this work, the rheological behavior of SBR self-degradable gels is evaluated by small angle oscillatory in-situ and ex-situ experiments. Additionally, relaxation time spectra are calculated where different relaxation modes are observed during the liquid-solid transition. The effects of polymer concentration, initiator and degradation agent concentration, temperature and solvent as transfer agent are evaluated. EXPERIMENTAL

Materials Cold emulsion polymerized styrene-butadiene rubber (SBR-8113) from ISP was used. It has the following values for weight and number average molecular weights, as determined by GPC: Mw= 247 ± 36 kg mol-1 and Mn= 171 ± 45 kg mol-1 (Vernáez et al 2015). Additives were removed by dissolving the polymer in toluene, following by precipitation with 1-propanol and further soxhlet solvent extraction.


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A 100% paraffinic mineral oil (Vassa® LP-90) was used as solvent. A mono-aromatic poly-alkylsubstituted solvent (Copesol®) was used as transfer agent. This solvent is a collateral product of the production of aromatic solvents such as benzene, toluene and xylene. From mass spectroscopy, Copesol® has a composition of 99.6 wt. % of mono-aromatic components: 33 wt. % of 1-ethyl-2-methyl benzene and 32 wt.% of 1,2,4 trimethyl benzene as the main components amongst other aromatic substances. All formulations were prepared with a polymer concentration above the critical overlap concentration c*, determined by simple shear experiments. It is important to point out that reaction is highly sensitive to oxygen and therefore all experiments were carried out in anaerobic conditions. Peroxides used were dicumyl peroxide (DCP), di-tert-butyl peroxide (DTP) and cumene hydroperoxide (HPC), all from Akzo Nobel. Dicumyl peroxide was used as crosslinker for low temperature formulations (105 to 120 °C) while di-tert-butyl peroxide as crosslinker for high temperature ones (138 to 149 °C). Cumene hydroperoxide was used as breaker for all formulations. Nitrogen 98% was used for in-situ reactions. Methods For ex-situ reactions, formulations were prepared by dissolving the polymer in the solvent for 24 hours at 50 °C with magnetic stirring. Some formulations contain Copesol® which is added in the solvent mixture before dissolving the polymer. Then, both peroxide and hydroperoxide were added to the solution and divided in different flasks. They were sealed in an oxygen free environment. Then, they were heated in the oven at the desired temperature. For each evaluation time, the flasks were removed from the oven and quickly cooled in cold water to slow down the reaction. Samples were taken directly from the flask to the rheometer for oscillatory shear experiments. A Physica MCR-301 Anton Paar Rheometer was used with PP-25 geometry. Linear viscoelastic ranges were determined for all samples at the higher and lower frequencies and at different gelation states. Frequency sweeps tests were performed at a deformation within the linear viscoelastic range. All evaluations were performed at 25 °C. Formulations which experimented syneresis or solvent expulsion were discarded from the study.


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For in-situ experiments, the formulation were added directly to a concentric cylinder geometry and placed in a Bohling Gemini II Rheometer from Malvern. A special assembly was implemented to ensure nitrogen purge and an oxygen free environment. Then, the geometry temperature is raised to the desired value and the oscillatory measurements were performed at different times during the reactions, always in the linear viscoelastic range. RESULTS AND DISCUSSIONS

In Figure 3.2, some of the most prominent radical reactions during SBR degradation are shown. Hydrogen abstraction is possible because radicals formed are stabilized either by double bond resonance or by the induction of delocalized -π-electrons of the aromatic ring. In crosslinking reactions with peroxides, some authors assume that all primary radicals created from homolytic decomposition of peroxides react with polymer chains to form a macro-radical (Mani et al. 2010). This assumption is valid, since usually primary peroxy radicals are reactive and hydrogen abstraction is normally faster than homolitic decomposition show in Figure 2. The use of spectroscopic techniques to evaluate the kinetics of the proposed self-degradable gel network was not possible because of sensitivity problems related to the small quantities of chemical changes involved to form a gel. Nevertheless, gels formed by polymer solutions experiment a liquid-solid transition due to interactions between polymer chains, which leads to a polymer network with marked changes in the mechanical properties of the system. The rheological behavior of a polymer solution crosslinked above its overlap concentration to form a chemical network is significantly different from gels formed by monomer reactions and also by cure reactions of bulk polymer systems. In the transition from solution to gel an infinitely large macromolecule is formed, which can only swell but not dissolve in a solvent (Richter, 2007). This work shows results for both ex-situ and in-situ experiments. Ex-situ experiments allow the evaluation of long times and higher temperatures, while in-situ experiments allow the study of crosslinking behavior at short times.


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Figure 3. 2. SBR radical reactions with peroxide Performing in-situ experiments implies the application of a frequency sweep during the crosslinking reaction, even though each frequency sweep can take up to several minutes. This introduces errors in the measurements because the reaction occurs in a transient state. Two in-situ experiments were done to check the difference between performing several frequency sweeps in time and the data was taken at a single frequency. Figure 3.3 shows as solid continuous lines, the experiments performed at a single arbitrary chosen frequency of 1.3 Hz (8.17 rad s-1) as a function of time, while the symbols represent the points taken from several frequency sweeps at the same frequency. In this work, it is assumed that no significant difference would be obtained if

ROOR 2RO

+R1 R2

R4R3

R1 R2

R4R3

+ ROH

R1 R2 R3 R4+

R1 R2

CH

R3

R4

Hydrogen abstraction

Addition

R1 R2+ R3 R4

R2

R4R1

R3

Crosslinking

R1

C R4

R2 R3 R3

R4

+R2

HC

R1

β-Scission

Homolytic decomposition

RO


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the same experiments had been performed and another single frequency has been chosen. The average time for each frequency sweep was 7 minutes, which is a significant time for some crosslinking experiments that results in unpredicted behavior of the experimental data. From these results, it is possible to assume that the continuous data at a single frequency are more accurate. In Figure 3.3, the behavior of the crosslinking process can be observed. At the beginning of the reaction, the instrument has not enough precision to measure the data. Addition and crosslinking reactions will progressively increase as the initiator decomposes with temperature. Once the polymer chains increase their molecular weight by reacting with each other above a certain value, a polymer network is created where effective entanglements increase the elasticity of the system. The formulation evaluated in Figure 3.3 did not contain hydroperoxide and then no degradation was observed during the evaluation time. Propagation steps in free-radical polymerizations are normally very fast, and termination by recombination of two macroradicals is diffusion controlled, which means that the termination constant is a function of macroradicals mobility. This phenomenon is called Trommsdorff effect or gel effect and can also be interpreted as if the termination rate constant depends on the viscosity of the system (Dusek and Dusková-Smrchková, 2000). Nevertheless, at long times, the slope of the curve decelerates, because the system reaches a topological limit, where radicals from polymer chains reduce their mobility making termination reactions less probable. Those radicals terminate either with primary radicals from the initiator or by β-scissions reactions.


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Figure 3. 3. In-situ crosslinking reaction for a 1.50 wt. % SBR solution with 0.20 wt. % of dicumylperoxide at 110 °C. Symbols represent the G’ and G’’ taken from frequency sweeps experiments at 1.3 Hz and continuous lines the essays performed at a constant frequency of 1.3Hz. The thermodynamics of the polymer network will determine the final rheological behavior of the gel. For instance, if crosslinker concentration is too high, the polymer network will have a higher crosslink density, reducing the swelling capacity of the network. For those cases, a shrinkage effect with an expulsion of the solvent from the polymer matrix may be observed, which is often called syneresis. Ex-situ experiments are less accurate because sampling and sample manipulation lead to experimental uncertainties. However, for large changes in the polymer network crosslink density, it is possible to quantify the viscoelastic behavior. Figure 3.4 shows the results for a self-degradable gel which increases its elasticity at the initial gel time, reaching a maximum at approximately 500 minutes, and then degradation proceeds and the consistency of the polymer network starts to decrease. For the case presented in Figure 3.4, only the points at 100 rad s-1 were taken. At the beginning of the reaction, polymer chains remain dissolved in the solvent matrix, and the system behaves more viscous than elastic. Homolitic decomposition of peroxide

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determines the initial gelation time and this being the rate determining step, it also determines the number of radicals produced as a function of time.

Figure 3. 4. Dynamic rheological moduli as a function of time for an ex-situ reaction at 100 rad/s for a solution containing 2.50 wt. % SBR, 2.75 wt. % of tert-butyl peroxide and 0.60 wt. % of cumene hydroperoxide Eventually, radical termination reactions increase the crosslink density and the storage modulus surpasses the loss modulus. The gel point is often referred to the moment where G’=G’’, however this point may vary from one frequency to another. Similar conclusions can be revealed from the tan δ (G’’/G’) curves with time. When the polymer network reaches its highest strength, topological constraints force macroradicals to undergo β-scission reactions in a degradation process that leads to low molecular weight chains, until a degradation time is reached at which G’ becomes lower than G’’. From this point onwards, a polymer solution is formed and the network can be considered completely destroyed. Figure 3.5 shows the experimental data of a frequency sweep performed at constant strain for a polymer solution at different reaction times. Although the G’ and G’’ curves indicate that some changes are occurring with reaction time, they do not reveal any information about structural changes. No differentiation between the contribution of physical entanglements and chemical crosslinks is observed. For example, a physical gel

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can be achieved by increasing polymer concentration, until G’ would be higher than G’’, even without any crosslinking reaction. Chemical gels are irreversible with bonds that have infinite lifetime, while in physical gels, van der Waals forces or hydrogen bonding can link polymer chains together, and those interactions have finite lifetime (Richter, 2007). The relaxation time spectrum can be described in terms of the generalized Maxwell model (Ferry, 1980) as a distribution of the times required for Maxwell’s elements to relax after a deformation has been applied to the sample. Equations 1 and 2 represent the discrete form of the generalized Maxwell model for the storage and loss moduli as a function of the frequencies ωj and relaxation times λi. Then, the relaxation time spectrum can be seen as a distribution of the Hi with their corresponding λi. After the liquid-solid transition, the system can never fully relax, and the equilibrium modulus is represented as Ge. (3.1)

(3.2)One of the major problems encountered in small angle dynamic rheology for these kinds of systems is the frequency range. For polymer fluids at low concentrations or low molecular weights before any association occurs, the linear viscoelastic range is found only for a short range of frequencies and some rheometers do not have enough sensibility to measure them. As chemical reactions or associations proceed, this frequency range extends and also the linear viscoelastic range could change (Vernáez and Müller, 2014).

++=

=2

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)('ij

ijn

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Figure 3. 5. Frequency sweeps for ex-situ experiments at different reaction times. Solid lines correspond to G’ values calculated from relaxation time spectra using Equation 1. Dotted lines correspond to G’’ calculated with Equation 2. Calculating the relaxation time spectra from the previous dynamic data (Figure 3.5) leads to a completely different appreciation of the system. Figure 3.6 shows the relaxation time spectra calculated by the method proposed by Vernáez and Müller (2014). The corresponding moduli calculated back from the spectra using equation (3.1) and (3.2) are presented in Figure 3.5 in continuous lines. Relaxation modes are related to the molecular mobility at different scales. Faster relaxations involve small scale motions of polymer chains, while slower relaxation modes are associated to large chain segments. Segmental relaxation is equivalent to the α-process in the dielectric spectrum (Wojnarowska and Knapik 2014). When interactions between polymer chains take place, as in high concentration polymer solutions, new long relaxation modes appear, related to intermolecular interactions, such as chain entanglements, which need more time to relax stress after deformation is applied to the system. The calculated relaxation time spectra exhibit different relaxation modes that are contributing to the final dynamic behavior of the samples in Figure 3.6. The first mode shown at short times is related to local modes, or simple entanglements. They relax

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faster and may not have any dependence neither with molecular weight nor to the liquid-solid transition. As it can be seen in the inset of Figure 3.6, the complex modulus evolves fast indicating the formation of a solid gel and then it decreases when degradation sets in until the material reaches liquid behavior.

Figure 3. 6. Relaxation time spectra for a reversible gel calculated at different reaction times in ex-situ experiments. Inner figure: Complex modulus at 10 rad s-1. For the initial polymer solution (curve labeled 0 min in Figure 3.6) and for the final degraded solution (curve labeled 4320 min in Figure 3.6), the local mode at short times can be observed in the spectra with maxima at similar times. For the other points in the solid state (at intermediate reaction times of 240 and 480 min), the local relaxation mode is shifted to higher times. This is a clear indication that local modes are affected by the macro-structural changes occurring in the system, because after crosslinking, local entanglements belong to a highly interconnected network and they can only relax slower than before crosslinking. Other relaxation modes are visible for the solid state in Figure 3.6. There are two medium scale relaxation modes that appear at intermediate times (that are longer than the time associated with the first local mode), representing larger structures or aggregates of physical entanglements. These intermediate modes are related to the relaxation of a larger mass of gel.

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Finally, a relaxation mode appears at the highest relaxation time and it diverges with time. This last mode is due to the chemical permanent crosslinks. Because of these chemical bonds, the structure will never be able to completely relax. This final relaxation mode is related to the maximum relaxation time explained by Winter and Mours (1997). Structural changes can be characterized by relaxation time spectra, including the liquid-solid transition. Effect of crosslinker Peroxide concentration is a dominant factor in controlling crosslink density. Radicals formed from diakyl peroxides are highly energized and are efficient at hydrogen abstraction (Dluzneski, 2001). Peroxide efficiency indicates the number of moles of crosslinks that can be formed from a mole of peroxide. Addition reactions can form covalent bonds between polymer chains and still let radicals available for crosslinking. For SBR high peroxide efficiency has been reported suggesting addition reactions are prevalent. The type of peroxide determines if SBR attack occurs by radical addition or by hydrogen abstraction. It has been reported for SBR that 80% of reactions occur by hydrogen abstraction and 20% by radical addition (Dluzneski, 2001). Primary radicals from peroxide decomposition disappear fast to form less reactive macroradicals in the polymer chains. Termination and addition constants determine the crosslinking rate. They decrease with time and with polymer concentration, as was demonstrated by Zhou (Zhou and Zhu, 1998). A set of formulations were prepared to study the effect of the peroxide concentration on the gel rheological behavior. As it was stated above, formulations studied for in-situ reactions were different from those studied in ex-situ ones, however, the peroxide concentration effect should be equivalent for both systems, as presented in Figure 3.7. The decrease in the initial gelling time is evident as the concentration of dicumyl peroxide increases, as well as the increase in reaction rate (Mani et al. 2010). The kinetic aspects of these reactions and the calculation of rate constants are out of the scope of this work.


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Figure 3. 7. In-situ reaction evaluated at 7.28 rad/s for a formulation of 1.50 wt. % polymer, and 110 °C Ex-situ experiments allow the collection of data for longer times and even higher temperatures, while in-situ experiments show the detailed evolution of complex modulus with time. It was not possible to performed in-situ reactions at high temperatures or times longer than 8 hours (480 minutes), because it implies keeping the nitrogen purge and rheometer running, compromising laboratory security and supplies. In-situ essays at higher temperatures were not performed due to solvent volatilization. Figure 3.8 shows general results for the ex-situ experiments through the behavior of the complex modulus as a function of reaction time. Reactions at higher concentration of crosslinker lead to shorter initial gelling times (less than 30 min versus 120 min for 2.25 and 2 wt. % of crosslinker respectively) and higher strengths (11.7 Pa, 11.2 Pa and 9.43 Pa for 3.00 wt. %, 2.25 wt. % and 2.00 wt. % of crosslinker respectively). After maximum strength, degradation rate of polymer network seems to be faster for high crosslinker concentration, at least until 2160 minutes. This effect can be explained by the diffusivity of macroradicals. For higher concentrations, a more rigid polymer network is formed with higher crosslink density, therefore termination and addition reactions became limited by the Trommsdorf effect. Radicals that are unavailable to

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terminate are prone to degrade by β-scission, then increasing peroxide concentration can also affect the overall balance among competing reactions.2

Figure 3. 8. Left: Complex modulus for a formulation with 2.5% SBR, 149 °C and 0.6 wt. % of breaker. Right: Relaxation time spectra for the tree formulations at 480 minutes. In a previous work (Vernáez et al. 2015), termination reactions were prevented by diluting the polymer solution, and degradation reactions of the polymer chains were observed by β-scission reactions. Relaxation time spectra were calculated for the three formulations at 480 minutes and are shown in Figure 3.8. All three formulations were measured after the liquid-solid transition and the longest relaxation mode corresponding to the chemical crosslinks are visible with a clear difference for 2.00 wt. % formulation. The local relaxation mode is shifted to higher times for the 2.00 wt. % peroxide formulation which is not consistent with the prior hypothesis of how crosslink density may affect local relaxation modes. As far as we are aware, a specific quantitative analysis of the relaxation time modes for this kind of spectra has been never achieved. Effect of polymer concentration The polymer concentration effect was also evaluated in terms of the rheological changes. For in-situ reactions, no breaker was added to the formulation. Figure 3.9 shows that tgi seems to be the same for the three reactions, although the slopes of the G* versus time curves are clearly proportional to polymer concentration. Addition and termination reactions should depend on polymer concentration, because the probability of polymer chains to meet each other is proportional to concentration.

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Nevertheless, gelling initiation time seems to depend only on the hydrogen abstraction step of primary radicals. From the kinetic point of view, this means that tgi should be a function of the number of radicals in the system in a specific time.

Figure 3. 9. In-situ evaluation of a formulation of 0.20 wt.% of crosslinker, at 110 °C and different polymer concentrations Figure 3.10 shows ex-situ evaluations for two formulations at different polymer concentration, at the same crosslinker and breaker concentrations. A difference between the two formulations arises from their maximum strength values (i.e., G* maximum value). A higher concentration of polymer implies more physically effective entanglements and also a higher probability of termination reactions. Another difference is the initial gelling time, tgi. For the same crosslinker concentration, both addition and termination reactions will be fast, as additionally, the topological limit will be reached in a short time. Nevertheless, because the formulations contain an amount of hydroperoxide as breaker, degradation reactions act also at different rates, retarding crosslinking reactions in different ways for both polymer concentrations.

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Figure 3. 10. Left: Ex-situ essays for two formulations with different polymer concentration and 2.75 wt. % of crosslinker and 0.6 wt. % of breaker. Right: Relaxation time spectra for both formulations at different reaction times Figure 3.10 (right) shows three relaxation time spectra. For the higher polymer concentrations at 240 minutes of reaction time, the contribution of local scale entanglements is higher than for the lower polymer concentration, even if the longest relaxation modes seem to be the same. This is clear evidence that the highest modulus measured is related not only to the covalent bonds created during crosslinking reactions, but also to a contribution of simple local entanglements. The relaxation spectrum for the formulation with 3.00 wt. % polymer at 480 minutes shows clearly a more significant contribution of covalent bonds at the longest relaxation mode. Effect of temperature The effect of temperature was evaluated for an in-situ formulation in the absence of breaker. In Figure 3.11, the effect of temperature is clear, reducing the initial gelling time and increasing the crosslinking reaction time. Temperature mainly affects the amount of radicals developed as a function of time. It is important to notice that in the absence of hydroperoxide, higher temperatures results in higher gel strength. This is in contrast with the results obtained for ex-situ reactions where hydroperoxide is in the formulation.

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Figure 3. 11. In-situ evaluation for different temperatures and the same formulation with 1.50 wt. % of SBR and 0.20 wt. % of dicumyl peroxide. Changing temperature affects both crosslinker and breaker in different proportion, due to their decomposition kinetics. As it was explained before, gel controlled degradation is achieved by a competition between crosslinking and degradation reactions. As shown in Figure 3.12 for an ex-situ evaluation, the maximum experimental temperature of 149 °C leads to a weak gel, because the degradation reaction takes place faster. For the minimum temperature, maximum gel strength is achieved; however the degradation time is longer. Also, the crosslinking reaction takes several minutes to reach the liquid-solid transition. For 145 °C, the maximum gel strength is about half that at 138 °C, and it is reached at 240 minutes, in contrast to the 480 minutes at 138 °C. Degradation time is also reduced by several minutes.

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Figure 3. 12. Left: Ex-situ experiments at three different temperatures for a formulation with 2.50 wt. % SBR, 2.75 wt. % crosslinker and 0.60 wt. % hydroperoxide. Right: corresponding relaxation time spectra at their maximum gel strength For application purposes it is important to know both initial gelation and degradation times determined at the temperature of the zone where the polymer solution will be injected. From the relaxation time spectra shown in Figure 3.12 (right), the difference in maximum gel strength for the three temperatures is evidenced. For 149 °C, the crosslinking reactions are expected to occur really fast, forming covalent bonds between chains, but because of the degradation, those bonds become quickly non-effective for the polymer network. That is why the longest relaxation mode is slightly visible in the spectrum and it does not divert with time. Effect of breaker concentration The breaker used in this work is a cumene hydroperoxide. In some applications, hydroperoxides can be used as a crosslinking agent, because they can form peroxyl radicals from homolytic decomposition. Vernáez et al. (2015) have demonstrated that hydroperoxide degrades SBR in dilute solution by a random scission mechanism. Degradation starts at the beginning of the reaction and degradation rate depends on polymer molecular weight. One possibility to explain this behavior is that hydroxyl radicals can abstract specific hydrogen atoms of the polymer chain, which leads to β-scission reactions. Hydroxyl radicals have clearly less steric hindrance and have more chance to abstract hydrogen from ternary carbons in the chains. The more available

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hydrogen from a ternary carbon is at the α-position because the radical formed is stabilized by the aromatic ring in the styrene repetitive unit of SBR. As the gel strength increases, degradation rate becomes higher, while termination and addition rates become lower by the gel effect and diffusion problems. The breaker concentration effect is better explained for the in-situ reactions presented in Figure 3.13. For the same crosslinker concentration of 15 wt.%, doubling breaker concentration leads to a higher initial gelation time and to a shift in the time needed to reach maximum gel strength. Nevertheless, the degradation times for both formulations are practically the same. This is another evidence of how the Trommsdorf effect has an influence in both crosslinking and degradation reaction. It can be assumed, that even if the concentration of breaker is low, a stiff gel can undergo degradation because termination reactions are reduced by mobility and then scission reactions are easier to occur. To give more evidence on this phenomenon, another formulation with half crosslinker agent has been studied. In this case, initial gelling time is shifted to longer times, the gel strength is reduced to less than half, but the degradation time is now increased several minutes. This result represents another evidence that competition reactions are directly related to the radical concentration in time.

Figure 3. 13. In-situ reversible crosslinking reaction for a formulation with 1.50 wt. % of SBR at 110 ºC and different crosslinker (DCP) and breaker (HPC) concentration 0 200 400 600 800 1000 1200

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Effect of transfer agent For practical applications, the reversibility of the gel needs to be adjusted to obtain the desired gelling time, gel strength and degradation time for a specific temperature. So far, the rheological data obtained for different temperatures and concentrations of crosslinker and breaker, indicate that the rheo-kinetic behavior of the system can be controlled by the formulation. However, for a specific temperature, the desired gel initiation and degradation times may not be achieved by only manipulating the crosslinker and breaker concentration. Controlling gelling times for free radical crosslinking reactions is achieved using scavengers that slow down the radical propagation reactions by protecting macroradicals. TEMPO scavengers are the most common reagents used for this purpose (Mani,2010). For industrial application aimed in this work, using TEMPO can increase the total operation cost significantly. In this work, a special solvent was selected to play a scavenger role during crosslinking and degradation reactions. Copesol® is an alkyl substituted monoaromatic solvent. Hydrogens in α position to the aromatic ring are suitable to be abstracted by primary radicals or even by macroradicals because they can stabilize radicals by induction effect of the aromatic pi cloud (Dannley and Zaremsky, 1998). This is introduced as an additional transfer step in the reaction, affecting both crosslinking and degradation. The effect of Copesol® in crosslinking reactions is clearly shown in Figure 3.14. A shift in the initial gelling time to higher values is proportional to Copesol® concentration. Besides, the termination rate is also affected by the addition of this solvent in the system. From the thermodynamic point of view, an aromatic solvent increases the solubility of the polymer chains and also their radii of gyration. Additionally, the critical concentration for coil-coil overlap (c*) should be shifted to lower values and chain mobility should be increased. The contribution from entanglements is expected to be


143 December 2015

higher with increased concentration of the aromatic solvent, as has been observed for higher polymer concentrations. The effect of this solvent quality on the degradation of SBR has been studied in a previous work (Vernáez et al. 2015). In that work, it was demonstrated that Copesol® retards degradation reactions. Primary radicals will prefer to abstract hydrogen from Copesol®, because its radicals are stabilized by induction of the delocalized electrons from the aromatic ring. Consumption of primary radicals, RO• and HO•, will affect both degradation and crosslinking reaction, although reaction rates are not equally affected. From in-situ reactions, it is possible to assume that primary radicals from the crosslinker are consumed by the solvent, retarding crosslinking reactions leading to low consistency gels with higher initial gelling times.

Figure 3. 14. In-situ evaluation of four formulation with 0.20 wt.% of Dicumyl peroxide, 1.50 wt.% of SBR at 110 °C and different Copesol® concentration in solvent 0 50 100 150 200 250

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Figure 3. 15. Relaxation time spectra for formulation 110 °C, 0.20 wt. % of Dicumyl peroxide, and 1.50 wt. % of SBR. The relaxation time spectra were calculated for four gels with similar complex moduli with different Copesol® concentrations as presented in Figure 3.15. Although all frequency sweeps were measured for the samples at times where the complex moduli were almost the same for all, the resulting spectra reveal large differences regarding relaxation modes. As expected, the contributions of faster relaxation modes are more significant for the samples with higher aromatic solvent concentration. For 0 and 5 wt. % of Copesol®, those contributions are overshadowed by the longest relaxation time modes. As concentration of aromatic solvent increased, the contribution of simple entanglements and medium scale network systems becomes more significant for relaxation behavior.

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Figure 3. 16. Ex-situ reaction for gel formulation of 2.65 wt. % of polymer, 3.00 wt. % of crosslinker, 0.10 wt. % of breaker at 149ºC and different Copesol® concentration

Figure 3. 17. Transfer reaction proposed for Copesol® and primary radicals For ex-situ experiments, reactions with different aromatic solvent concentrations were also evaluated to study the reversibility in the mechanical properties of the gels, as shown in Figure 3.16. For these particular formulations, it is possible to observe changes in reversible behavior, in gel strength, initial gel time and degradation. The rheo-kinetic of samples is affected by introducing a new radical transfer step in the reaction, either by transferring radical to the solvent or termination by coupling with a radical from solvent molecules as is schematized in Figure 3.17.

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Conclusions

The rheological characterization of self-degradable gels can be performed either by in-situ or ex-situ measurements. Polymers in solution experiment a liquid-solid transition and structural changes to form a polymer network with physical entanglements and chemical crosslinks. Adding hydroperoxide to the initial formulation generates hydroxyl radicals which degrade the polymer chains as they simultaneously crosslink. The kinetics of crosslinking and degradation reactions are dependent on the living radicals at specific moments in time. The Trommsdorf effect has an influence in radical diffusion and constraints termination and addition reactions, while degradation is increased. Structural changes in polymer networks can be deduced from calculated relaxation time spectra, where several relaxation modes appear showing contributions of local simple entanglements and chemical covalent bonds to the final rheological behavior. Acknowledgements


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MANI, S., et al. Rheological modelling of the free-radical crosslinking of PDMS rubber in the presence of TEMPO nitroxide. Polymer. 2010, vol. 51, pp. 3918–3925. RICHTER, S. Recent Gelation Studies on Irreversible and Reversible Systems with Dynamic Light Scattering and Rheology - A Concise Summary. Macromolecular Chemistry and Physics. 2007, vol. 208, pp. 1495–1502. ROTHS, T., et al. Determination of the relaxation time spectrum from dynamic moduli using an edge preserving regularization method. Rheological Acta, 2000, vol. 39, pp. 163–173. STADLER, Florian J. On the Usefulness of Rheological Spectra—a Critical Discussion. Rheologica Acta, 2013, vol. 52, no. 1, pp. 85-89. STADLER, Florian J.; and BAILLY, Christian. A New Method for the Calculation of Continuous Relaxation Spectra from Dynamic-Mechanical Data. Rheological Acta, 2009, vol. 48, pp. 33-49. VENTRESCA, M. L., FERNÁNDEZ, I. and NAVARRO-PEREZ, G. Reversible Gelling systems and methods using same during well treatments. US Patent 7,638,467 B2 filed 3 Aug. 2004, and issued 29 Dec 2009. VERNÁEZ, O., et al. Degradation of styrene butadiene rubber ( SBR ) in anaerobic conditions. Polymer Degradation and Stability. 2015, Vol. 111, pp. 159–168. VERNÁEZ, Oscar; and MÜLLER, Alejandro J. Relaxation Time Spectra from Short Frequency Range Small-Angle Dynamic Rheometry. Rheologica Acta, 2014, vol. 53, no. 5-6, pp. 385-399. WANG, Y., LUE, A. and ZHANG, L. Rheological behavior of waterborne polyurethane starch aqueous dispersions during cure. Polymer. 2009, vol. 50, pp. 5474–5481. WINTER, Horst H.; and MOURS, Marian. Neutron spin echo spectroscopy viscoelasticity rheology Springer, 1997. Rheology of Polymers Near Liquid-Solid Transitions, pp. 165-234. WOJNAROWSKA, Z. AND KNAPIK, J. New insight into relaxation dynamics of an epoxy hydroxy functionalized polybutadiene from dielectric and mechanical spectroscopy studies. Colloids and Polym. Science. 2014, vol. 292, pp.1853–1862. YOUNG, R. J. & Lovell, P. A. Introduction to Polymers. Second ed. London: Champan&Halls, 1991.


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ZHOU, W. and ZHU, S. ESR Study of Peroxide-Induced Cross-Linking of High Density Polyethylene. Macromolecules. 1998, vol. 31, pp. 4335–4341.

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CHAPTER IV

Degradation of StyreneButadiene

Rubber (SBR) in Anaerobic Conditions

Oscar Vernáez, Sylvie Dagreou, Bruno Grassl, A.J Müller

Doctoral student contribution: In this chapter, the student performed all the experiments, along with the analysis using population balance equations. He also wrote the paper, which was submitted and accepted in Polymer Degradation and Stability. He was the corresponding author of the publication.

Chapter IV: Degradation of Styrene Butadiene Rubber (SBR) in Anaerobic Conditions

151 December 2015

ABSTRACT

In this work, the degradation kinetics of styrene butadiene rubber (SBR) in solution was studied in anaerobic conditions. Degradation reactions in the presence of cumene hydroperoxide at different concentrations (0.20, 0.28, 0.32, 0.50, 0.60 % in weight), temperatures (60, 75, 85, 100 and 120 ºC) and aromatic solvent concentrations (10 and 20 %) were performed. The fragmentation rates of polymer chains, which define the degradation kinetics, were calculated from the change in molecular weight distribution with time. The degradation was performed in a reactor with anaerobic conditions and the characterization was performed by multiangle light scattering coupled to size exclusion chromatography (SEC-MALS). Using population balance equations, it was possible to calculate the kinetic constants for thermal and thermooxidative degradation. Analysis of the results led to the conclusion that random scission of polymer chains produced by macroradicals formed by hydrogen abstraction constituted the predominant SBR degradation mechanism. Adding alkylbenzene as a transfer agent significantly reduced the degradation.


152 December 2015

RESUMEN

En este trabajo se estudió la cinética de degradación del caucho estireno butadieno (SBR) en solución bajo condiciones anaeróbicas. Se realizaron las reacciones de degradación en presencia del hidroperóxido de cumeno a diferentes concentraciones (0,20; 0,28; 0,32; 0,50; 0,60 % en peso), diferentes temperaturas (60, 75, 85, 100 y 120 ºC) y diferentes concentraciones de solvente aromático (10 y 20 %). A partir del cambio de la distribución de peso molecular en el tiempo se calcularon las velocidades de fragmentación de las cadenas poliméricas que definen la cinética de degradación. La degradación se realizó en un reactor bajo condiciones anaeróbicas y la caracterización se llevó a cabo utilizando un dispersor de luz multiánculo acoplado con un sistema de cromatografía por exclusión de tamaño (SEC-MALS). Utilizando las ecuaciones de balance poblacional fue posible calcular las constantes cinéticas para la degradación térmica y termo-oxidativa. Luego del análisis de los resultados se concluyó que el mecanismo predominante de degradación del SBR era la escisión al azar de las cadenas poliméricas producto de los macroradicales que se forman por la reacción de abstracción de hidrógenos. Añadiendo un alquilbenceno como agente de transferencia se reduce significativamente la degradación.


153 December 2015

RÉSUMÉ

Dans ce travail, la cinétique de dégradation du styrène butadiène (SBR) en solution a été étudiée en conditions anaérobies. Les réactions de dégradation en présence d’hydroperoxide de cumène ont été réalisées pour différentes concentrations en d’hydroperoxide de cumène (0.20, 0.28, 0.32, 0.50, 0.60 % en masse), différentes températures (60, 75, 85, 100 et 120 ºC) et différentes concentrations en solvant aromatique (10 et 20 %). A partir des changements dans la distribution des masses moléculaires, les taux de fragmentation des chaînes de polymère, qui définissent la cinétique de dégradation, ont été calculés. La dégradation a été mise en œuvre dans un réacteur anaérobie, et les chaines polymères ont été caractérisées en diffusion de la lumière multi-angles couplée avec la chromatographie d’exclusion stérique (SEC-MALS). En utilisant les équations de bilan de population, nous avons pu calculer les constantes cinétiques de dégradation thermique et thermo-oxydative. L’analyse des résultats conduit à la conclusion que la scission aléatoire des chaines de polymère due à la formation de macro-radicaux par abstraction d’hydrogène est le mécanique prédominant de la dégradation de SBR. L’ajout d’alkylbenzène en tant qu’agent de transfert réduit la dégradation de façon significative.


154 December 2015

Introduction

Physical properties of polymers depend to a large degree on molar mass, its distribution and molecular architecture, therefore, chain scission and crosslinking play an important role in polymer performance. For this reason, it is important not only to evaluate the chemical changes in a polymer reaction, but also the changes in molar mass distribution (Chiantore, Tripodi, Sarmoria, Vallés 2001). Polydienes degradation occurs at specific sites because double bonds in the main chain act as reactive points. A series of hydrogen transfer reactions may occur in polydienes when they are heated in the absence of oxygen. Transfer of allylic hydrogen can lead to chain scission. Pendant vinylene groups promote the lability of allylic hydrogen, producing stable radicals, leading to β scission processes or chain recombination(Chiantore, Tripodi, Sarmoria, Vallés 2001; Jiang, Levchik, Levchik, Wilkie 1999). The scheme of the β scission for polybutadiene (PB) is shown in Figure 4.1a and 4.1b. In the presence of peroxides, degradation of these polymers occurs throughout a mechanism of abstraction of the allylic hydrogen in the main chain, which may result in either a β scission or a crosslinking reaction from the recombination or termination reaction at high concentration. Thermal degradation of PB has been described with the reaction scheme shown in Figure 4.1b(Jiang, Levchik, Levchik, Wilkie 1999). Likewise, for polystyrene (PS) degradation, a free radical mechanism has been proposed where the radical generated from the hydrogen abstraction of the tertiary carbon can be stabilized by induction effects of aromatic rings and can then undergo chain scission as shown in Figure 4.1c (Gordon Cameron, Meyer, McWalter 1978).

Chapt

Fi199Thesthey systePeroreact

ter IV: Degrada)

b)

c)

igure 4. 1. a99) b) ReactLevchse scission rcan be conem and leadxides and tive primar

dation of Styr

a) β Scissiontion schemhik, Wilkie 1Polystyrenreactions dnsidered asd to low mohydroperory alcoxy ra

rene Butadien

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n of PB frome proposed1999) c) Mene(Gordon o not implys chain reaolecular weoxide homadicals, wh

ne Rubber (SB

155 ecember 20

m allylic rad for thermaechanism fCameron, M y the terminactions thateight produolitic decohich are sta

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156 December 2015

from the polymer chains to form more stable tertiary or allylic radicals(Kim, McCoy 2000; Madras, Chattopadhyay 2001). Polymer degradation processes may be described by population balance equations (PBE), which can model the changes in molar mass distribution as a consequence of polymer chain fragmentation. In this work, the kinetics aspects of the anaerobic degradation of a styrene butadiene rubber (SBR) dissolved in a paraffinic mineral oil as a function of time is evaluated. The parameters studied are: temperature, concentration of an organic hydroperoxide, and concentration of a radical transfer agent. The molar mass distribution is obtained by gel permeation chromatography (GPC) coupled with multiangle light scattering (SEC-MALS). The fragmentation kinetic is analyzed by population balance equation (PBE) theory. Model description

One of the advantages of using PBE is the possibility of including phenomenological parameters that affect the kinetics of degradation reactions. It is also a way to handle continuous mixtures of particles of the same chemical features, and analyzing continuous reactions in one single integro-differential equation If x is the molar mass of a polymer chain, the weight molar mass distribution is given by xf1(x, t), while the number molar mass distribution is f1(x, t). The molar concentration of the polymer in a molar mass range of (x,x+dx) is f1(x, t)dx. Thus, the total number of particles in a specific time t, can be calculated as: = dxtxftN ),()( 1 (4. 1)

Chapt FigurConst-t0 is

The n

ThenmolamoleThe frepre

ter IV: Degradre 4.2 showsidering a sus given by t

Fn-th momen

n, the zero ar mass dises of polymfirst momeesented by

dation of Styrws two deubset of thethe shaded

Figure 4. 2. Mnts of the d

moment istribution, aer chains. ent is the to(4. 4):

rene Butadien

D

ensity functe x domainregion, def

Molar massdistribution

s given by tand represotal mass o

txfdtd b

a ),(1

∞

=0

xnnμ

ne Rubber (SB

157 ecember 20

tions, one , [a,b], the fined as (Ra

s distributin can be als

the area beents the tof the polym

dtdNdx =)

1 ),( dxtxf

BR) in Anaero

015

at time t=change in namkrishna 2

on at two do defined a

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urve defineconcentratis concentra

ns e other at molecules i

mes

ed by the nion, or numation, and m

time t. in time (4. 2)

(4. 3)number mber of may be


158 December 2015

(4. 4)Assuming mass conservation during the reaction, the first moment should be constant in time. From distributions moments it is possible to determine other important characterization parameters, such as the number average molecular weight Mn = μ1/μ0, the weight average molecular weight Mw =μ2/μ1; and the polydispersity index Mw/Mn =μ2μ0/μ12. The weight molar mass distribution xf1(x, t) can be expressed as a function of Mn and Mw through a gamma function type as the Schulz-Zimm distribution function(Tobita 1995):

(4. 5) Where κ is given by:

(4. 6)In fragmentation or degradation of polymer chains, the PBE for pure fractionation reactions can be defined as(Ramkrishna 2000):

(4. 7) Where b(x’) is the breakage rate, defined as the fraction of particles of size x’ that break in a specific time unit. v(x’) is the average number of particles formed from the

∞

=0

11 ),( dxtxxfμ

tn

k

nn Mx

Mx

Mtxxf

−

Γ

= κκ

κ κ

exp)(

),(1

nw

n

MMM−

=κ

∞

−=∂

∂

x

txfxbdxtxfxxPxbxvt

txf ),()(`),'()'|()'()'(),(11

1


159 December 2015

breakup of particle x’. Assuming one breakup at a time for a single particle, i.e., binary breakup, the value of this function would be v(x’)=2. P(x|x’,t) is defined as the probability density of particles of size x which come from the breakup of particles of size x’ at time t. It is also called stoichiometric Kernel, and represents the distribution of sizes of chains generated from the breakup of particles of size x’. When the breakup occurs randomly in the chain, the Kernel would be P(x,x’)=1/x’ as described by some authors(Kim, McCoy 2000). There are different methods to solve the integro-differential equation that describe the PBE. Ramkrishna(Ramkrishna 2000) describes several methods that may be applied depending on the characteristics of each system. For pure fragmentation, the moments of the density function may be used as a tool for solving PBE. Some authors (Ramkrishna 2000) have described the moment solution assuming no particle growth and considering binary fragmentation ν(x)=2. The reaction mechanism for thermoxidative degradation of polymer with its different kinetics constant is summarized in the next scheme of equations. Thermal degradation:

(4. 8) Initiator decomposition:

(4. 9) Hydrogen abstraction by the primary radicals: (4. 10)

)'()'()( 0 xxPxPxP k −+⎯→⎯

OHROROOH k •• +⎯→⎯ 1

)()( 2 xRROHxPRO k •• +⎯→⎯+


160 December 2015

(4. 11) β scission of macroradicals where it is important to highlight that the macroradical R(x’) is a terminal radical (Chodak, Zimanyova 1984; Karmore, Madras 2002a):

(4. 12)Chain radical transfer:

(4. 13) where x and x’ represent two general molecular weights with ( ) From these equations the PBE can be constructed through equation (4.7). Naming p(x,t) the density function of polymer in a specific time t; the PBE for thermal degradation can be represented by:

(4. 14)Now, for degradation reaction with the added degradation agent or hydroperoxide, where p(x,t) and r(x,t) express the polymer and macro-radical concentration in time t respectively, the PBE for both macro-radical and polymer chains can be represented as:

(4. 15)

)()( 23 xROHxPHO k •• +→⎯→⎯+

)'()'()( 4 xxPxRxR k −+⎯→⎯ ••

)'()'(9

9

xPxRb

a

k

k

⎯⎯←⎯→⎯•

xx <'

∞

+−=∂

∂

x

dxxxPtxpxktxpxkt

txp ')'|(),'()(2),()(),(00

++−−−=∂

∂•• ),()(),()(),()(),()(),(

9932 txrxktxpxktxptxCktxptxCkt

txpabHORO


161 December 2015

(4. 16)

Experimental

Materials The polymer employed in this work was a cold polymerized Styrene butadiene rubber SBR from ISP (SBR-8113). This polymer was purified by solvent sohxlet extraction with toluene and then precipitated with 1-propanol and dried in a vacuum oven at 40ºC. The mineral oil, Vassa LP-90 was used as a solvent. It is a 100% paraffinic solvent composed of C6-C22 molecules. This solvent was filtrated twice with a 0.2 μm Millipore®. An aromatic solvent was also used, named Copesol®, from Pequiven S.A. This solvent is a mixture of poly-substituted monoaromatic molecules of the alkybenzene type. Cumene hydroperoxide 80% (Trigonox K-80) from Akzo Nobel was also used with an activation energy (Ea) of 132.56 kJ/mol and an Arrhenius constant (A) of 1.15x 1012 reported by the supplier. Nitrogen 98% was used to purge the reaction.

''),'(2),('

1),'(' 004 ∞∞

+−xx

dxtxpktxxpkdxx

txrxk

+−++=∂

∂•• ),(),(),()(),()(),(

9932 txxrktxxpktxptxCktxptxCkt

txrbaHORO

),(''

1),'(' 44 txxrkdxx

txrxkx

−∞


162 December 2015

Degradation reaction The degradation assembly consists on a three-neck flask with a jacket connected to a recirculation bath for maximum temperature control and magnetic stirring. A controlled nitrogen in-out system was installed with a septum plug for needle sampling. The purified SBR was dissolved in the mineral oil at 40ºC and at a concentration of 2 wt% for at least 48 hours prior to the degradation reaction. Then the nitrogen purge system was mounted and the temperature was raised to the desired value. Time zero was considered when the recirculation bath reached the defined temperature. The hydroperoxide was added right before rising the temperature. Samples of 1 cm3 were taken during the reaction at different times and were dissolved in the elution solvent (in a 1:1 volume ratio) prior to the injection in the SEC-MALS. The degradation was performed without hydroperoxide at temperatures of 60°, 100° and 120 °C. For 100 °C, three different concentrations of cumene hydroperoxide were employed: 1.3x10-3 M (0.50 wt%); 8.41x10-4 M (0.32 wt%) and 7.36x10-4 M (0.28 wt%). For 120 °C, two hydroperoxide concentrations were studied: 1.6x10-3 M (0.20 wt%) and 5.26x10-4 M (0.60wt %). For solutions containing 1.3x10-3 M (0.50 wt%) cumene hydroperoxide, three temperatures were tested: 75, 85 and 100 °C. Finally SBR was dissolved in three mixtures with aromatic solvent, Vassa:Copesol (9:1; 8:2 and 7:3). The hydroperoxide concentration was kept at 0.50 wt% and the temperature was fixed at 100 ºC. Caracterization procedures A size exclusion chromatograph (SEC) coupled with a multiangle light scattering (MALS) detector and a refractometer was used to determine the molar mass distribution. Gel permeation chromatography (GPC) columns were used as SEC device,


163 December 2015

with Mixed C and Mixed D and linear columns from Variant. An isocratic Agilent 1200 pump was used. The elution solvent was either COPESOL or filtrated HPLC grade toluene. The MALS system employed was a DAWN EOS (Wyatt Technology) coupled with a dn/dc refractometer Optilab-Rex (Wyatt Technology) to determine concentration. All measurements were carried out at 25ºC and data were processed by ASTRA V software from Wyatt Technology. The dn/dc value for the SBR in the aromatic solvent was 0.0478 ± 0.0009 mL/g and 0.0456 ± 0.0011 mL/g determined in batch. This value was not affected by the mixture of the polymer with the mineral oil, because the chromatography columns allow the separation of the polymer chains from the mineral oil in the sample. The oil signal eluded at higher times. No difference was observed when the elution solvent was Copesol or toluene. Experimental results

Reproducibility and repeatability SBR used in this study was a heterogeneous industrially processed polymer. Before any degradation was induced, a solution of SBR was prepared and purified. At time zero, where no temperature had been applied, no differences in molecular mass should be appreciated in between different samples of the same SBR batch employed here. Nevertheless, Figure 4.3 shows the large differences found for 25 similar SBR samples at time zero. These tests include two or three runs of the same sample, i.e., reproducibility and repeatability are considered in the same figure. The initial average Mw for all samples at initial time is shown in the inner table of Figure 4.3. The high values of the standard deviation represent the heterogeneity of the sample. These errors have to be taken into account during the degradation analysis. It is important to point out that SEC-MALS technique is an absolute method to determine Mw, and then the conformational changes due to temperature or solvent interaction with polymer


164 December 2015

chain do not affect the results. Neither the concentration changes because of non-soluble fraction nor the polymer retention in the columns would change the results, because the equipment is able to detect absolute concentration. This is why those differences are assumed to be just a matter of sample heterogeneity.

Figure 4. 3. Reproducibility and repeatability tests for SBR samples in solution at initial time (before degradation). The middle solid line represents the average Mw and dash lines represent one standard deviation Degradation results Figure 4.4 shows an example of the change in the molecular weight distribution with time during degradation reaction with hydroperoxide and temperature. It can be seen that the molecular weight decreases asymptotically until it reaches a minimum value (see the inset in Figure 4.4). This decrease is important in comparison to the values of the standard deviation observed. These results suggest that degradation process might be proportional to molecular weight as predicted. From the distribution it is possible to observe the decrease of polydispersity. This behavior is opposite to what it could be expected for random scission. During degradation, all molecules tend to a

0 5 10 15 20 25

100

150

200

250

300

350

400

Mw

(kg/

mol

)

run

Mw (Kg/mol)

Mn (Kg/mol)

Average 246.854 170.747 Standard deviation 35.832 45.358

% Std 14.5 26.6


165 December 2015

specific molar mass and then the distribution broadness also decreases. This can be explained by the asymptotic behavior of the molecular weight with time. The degradation reaction tends to stop after reaching some minimum molecular weight value. Some authors have numerically simulated this behavior as single frequency distribution function dynamics, considering that the degradation stops after some minimum molecular weight and totally random kernel. The initial frequency distribution function decreases exponentially, generating a product peak which eventually tends to a Dirac delta function for long times(McCoy, Benjamin J. Wang 1994).

Figure 4. 4. Distribution of molecular weights for a degradation reaction of a SBR sample in solution with 0.6 % of hydroperoxide at 120 °C. The inset shows how the Mw changes with reaction time Figure 4.5 shows the results of three thermal degradation experiments without the use of hydroperoxide at different temperatures. It can be seen that for 60 and 100 °C there is a slight increase in molecular weight that falls within the standard deviation shown in Figure 4.3. There was no evidence of Mw increment during the chromatography results. From the practical point of view, in this work, this increase will be considered negligible. On the other hand, for 120 °C, the reduction in molecular

103 104 105 106 107

0.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

0.0 5.0x104 1.0x105

0

1x105

2x105

3x105

Mw

(g/m

ol)

time (s)

Diff

eren

tial w

eigh

t fra

ctio

n (m

ol/g

)

molar mass (g/mol)

t=0min t=60min t=120min t=215min t=340min t=525min t=1425min


166 December 2015

weight is significant. Thus, there is a critical temperature at which thermal degradation begins to be noticed. Some authors suggest chain scissions will take place if the heat exceeds the dissociation energies of chemical bonds(Schnabel 1981).

Figure 4. 5. Thermal degradation experiments of SBR in solution for three different temperatures, without the use of hydroperoxide. The change in Mw was normalized with respect to the initial Mw of the sample Degradation reactions with cumene hydroperoxide were carried out at different temperatures in an anaerobic environment, at a fixed concentration of cumene hydroperoxide (0.5%). Figure 4.6 shows the degradation of SBR polymer chains in the presence of a constant concentration of hydroperoxide. For 75 °C, degradation is negligible. Changes in molecular weight start to be significant at 85 °C; at this temperature, the Mw value reaches 50% of its initial magnitude in 25,000 seconds. At 100 °C, the polymer chains reach an asymptotic behavior at approximately 30 % of the initial weight average molecular weight after 10,000 seconds.

0 20000 40000 60000 80000 100000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

60 ºC 100 ºC 120 ºC

Mw

/Mw

0

time (s)


167 December 2015

Figure 4. 6. Degradation of SBR in solutions with 0.50 % of cumene hydroperoxide at three different temperatures. Degradation reactions were also performed at different concentrations of cumene hydroperoxide, at two different temperatures. At 100 ºC three different concentrations of the hydroperoxide were evaluated, while for 120 ºC only two concentrations were explored. Figure 4.7 shows the degradation behavior of SBR as a function of time for all hydroperoxide concentrations. Once more, an asymptotic behavior was observed for the molecular weight reduction as a function of time. The degradation extend was large, since the samples exhibited Mw values close to less than 10 % of the molecular weight of the undegraded samples. It can be also seen that no significant differences in degradation can be appreciated for the different hydroperoxide concentrations (within the errors involved in the measurements). Nevertheless, it is clear that at temperatures of 100 ºC and 120 ºC, the degradation kinetic is accelerated by the addition of hydroperoxide.

0 5000 10000 15000 20000 25000 30000

0.0

0.5

1.0

1.5 100°C 0.5% HPC 75°C 0.5% HPC 85°C 0.5% HPC

Mw

/Mw

0

time (s)


168 December 2015

Figure 4. 7. SBR degradation at different hydroperoxide concentration. a): 100 ºC. b): 120ºC The effect of an alkylbenzene, as a radical transfer agent, was also evaluated at different concentrations and in the presence of the hydroperoxide. If the degradation reaction depends on hydrogen abstraction from a radical, the presence of a transfer agent should retard the degradation reaction by retaining and stabilizing radicals. Figure 4.8 shows how the addition of different concentrations of the transfer agent retards significantly the degradation reaction. This is an important result in order to be able to control the degradation reaction.

0 20000 40000 60000 80000 100000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0% HPC 0.28% HPC 0.32 % HPC 0.50% HPC

Mw

/Mw

0

time (s)

a)

0 20000 40000 60000 80000 100000

0.0

0.2

0.4

0.6

0.8

1.0 0 % HPC 0.20 % HPC 0.60 % HPC

Mw

/Mw

0

time (s)

b)


169 December 2015

Figure 4. 8. Degradation of SBR in solution with different concentrations of an aromatic solvent (Alkylbenzene or Copesol). The experiments were performed at 100 ºC and with a concentration of hydroperoxide of 0.5% Discussion of the results

By the use of spectroscopic methods, it was not possible to follow the degradation reaction through the quantification of a chemical change in the sample. Hydroperoxide consumption was measured by titration and no difference was observed as compared to the homolitic decomposition of the hydroperoxide itself. The appearance or disappearance of functional groups was not enough to be quantified by available methods in our laboratory. The balance between peroxide, peroxy radicals, and polymer molecular weight should rule the time evolution of the degradation process. The hydroperoxide decomposition implies the formation of two radicals of different kinds, a hydroxy radical and a cumyloxy radical. The decomposition rate is of the Arrhenius type. In terms of the molar concentration of peroxide, c, the dissociation rate is given by kic where the coefficient ki is governed by a first order kinetics, ki=Aexp(-Ea/RT). This constant was calculated for different temperatures as shown in

0 5000 10000 15000 20000 25000 30000 35000 40000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Pure alyphatic solvent 10 % Alkylbenzene 20% Alkylbenzene

Mw

/Mw

0

time (s)


170 December 2015

Table 1 using the values of the activation energy and the value of A, given by the supplier. Regarding the kinetic study of polymer degradation, many authors (Mccoy 1996; Madras, Smith, McCoy 1996; McCoy, Madras 1997; Kodera, McCoy 1997; Madras, McCoy 1997; Madras, Smith, McCoy 1997; Madras, McCoy 1998; Sterling, McCoy 2001; Kim, McCoy 2000; Karmore, Madras 2002a, 2002b) have generated mathematical functions to describe changes in molecular weight distribution through population balance equations, considering in some cases the initiation, propagation, transfer, β scission and termination reactions. To achieve this it is necessary to neglect some phenomena, such as density changes, crosslinking or aggregation, diffusion of species or Trommdorf effect. Table 1. Kinetic constants for Cumene hydroperoxide decomposition for different temperatures. Temperature (°C) ki (1/s) 75 1.49x10-8 85 5.33x10-8 95 1.79x10-7 100 3.19x10-7 105 5.62x10-7 120 2.81x10-6 The integro-differential relationships for the PBE are solved in terms of molecular weight using two main approximations: the long chain approximation (LCA) where initiation and termination reactions are infrequent events, thus are neglected in comparison to hydrogen abstraction reactions. The other approximation is the quasi-steady state approximation (QSSA), where the change of radical concentration with time is too small as compared to the change of other components and could also be neglected (Smagala, Mccoy 2003) . The changes in the molecular weight also show that the degradation process is not due to a depolymerization process, but a consequence of some chain fragmentation.


171 December 2015

For a depolimerization process, no change in molecular weight distribution should be detected and the decrease in the average molecular weight would be smoother. These kinds of fragmentations represented by random scissions should be described by PBE employing the stoichiometric Kernel as the inverse function, as suggested in the literature (Mccoy 1996), where P(x|x’)=1/x’. High molecular weight chains degrade faster than short chains and this is related to the number of hydrogen atoms available for abstraction, i.e., the larger the number of hydrogen atoms present in the chain, the higher the probability of a hydrogen abstraction reaction. This behavior has been predicted by some authors (Madras, McCoy 2007; Mccoy 1996; Madras, Chung, Smith, McCoy 1997) for hydrogen abstraction degradation mechanisms. From PBE, this behavior implies that function b(x) should be proportional to the molecular weight. The break function has been reported as linearly dependent on the molecular weight b(x)=kx or even exponentially b(x)=kxr (Mccoy 1996). Studies on the molecular weight influence on degradation indicate that random-scission rate coefficient dependence on molecular weight is linear for short chains, but second order for longer chains(Madras, Chung, Smith, McCoy 1997). It may also depend on other variables, such as temperature, concentration or reaction type. A simplification frequently used in PBE is binary fractionation v(x)=2, which assumes that one chain can only suffer a single fragmentation at a time, i.e., non simultaneous fragmentation for the same molecule is considered. The general degradation reaction can be divided into different reactions assuming some mechanism. After decomposition of the ROOH hydroperoxide, the resulting OH hydroxy and RO cumyloxy radicals, which are not stable, abstract hydrogens from the polymer chain P(x), generating polymer radicals or macroradicals R(x), where x represents the molecular weight as a continuous variable.


172 December 2015

PBE for thermal degradation Assuming linear dependence of the degradation reaction with the molecular weight, PBE for (4.14) in the case of thermal degradation can be written as: (4. 17)

Applying the moment operation, i.e., multiplying by xn and integration in the semi-infinite interval, the following integro-differential equation is obtained:

(4. 18)

Rearranging and knowing that :

(4. 19)

For the zero moment, n=0 substituting in the equation 4.19:

(4. 20)

Then Ns can be defined as:

∞

+−=∂

∂

x

dxtxpktxxpkt

txp '),'(2),(),(00

'),'(2),(),(

000

10 dxdxxtxpktxpk

ttxp x

nnn

+−=

∂∂

∞

+

∞

=0

),(),( dxtxpxtxp nn

)1()1(),(),(

),'(1

2),(),(

10

1010

+−−=

∂∂

++−=

∂∂

+

++

nntxpk

ttxp

txpn

ktxpkt

txp

nn

nnn

tktp

tptptp

ttpktptp

01

0

1

0

10

00

)()0(

)()(

)()0()(

==−

==−


173 December 2015

(4. 21) Figure 4.9 shows the results of applying these equations to the thermal degradation at different temperatures. It can be seen that for 100 ºC and 60 ºC the slopes are negative and very close to zero. This behavior is related to that shown in Figure 9. For this work, these constants can be considered zero and it can be assumed that the negative values of the slope are related with experimental errors. On the other hand for 120 ºC, Figure 4.9 shows that the value of the slope is significant, then, the thermal degradation reaction should be taken into account with a value of k0= 2.79x10-10s-1.

Figure 4. 9. Calculation of the thermal degradation kinetic constants by PBE for three different temperatures.

tkMtM

Nsnn

00

1)(

1 =−=

0 20000 40000 60000 80000 100000

0.0

5.0x10-6

1.0x10-5

1.5x10-5

2.0x10-5

2.5x10-5

tkMtM

N snn

00

1)(

1 =−=

60 °C 100 °C 120 °C

Ns

time (s)

k0120 °C = 2.79x10-10

k060 °C< k0

100ºC < 0


174 December 2015

PBE for degradation with hydroperoxide For degradation reaction with the hydroperoxide, in equations (4.15) and (4.16) the hydrogen abstraction from primary radicals (CRO.) and (CHO.) could be considered as a single process of hydrogen abstraction from C(t) with a kinetic constant k23. Assuming the stoichiometric Kernel as the inverse function P(x|x’), =1/x’ and applying the moment operation to equations (4.15)and (4.16): (4. 22)

(4. 23) (4. 24)

On the other hand, for the hydroperoxide decomposition, the concentration of hydroperoxide with time is given by: (4. 25)

The PBE expressed in terms of the primary radical concentration are given by:

'),'(2),(

'1),'('),(),(),()(),(

000

10

04

0

19

0

19

0

123

dxdxxtxpktxpk

dxx

txrxxktxrxktxpxktxpxtCkt

txp

xnn

x

nna

nb

nn

+−

++−−=∂

∂

∞

+

∞∞∞+

∞+

∞+

)1()1(),(

)(1

),(),(),()(),(

10

1419

19

123

+−−

+++−−=

∂∂

+

++++

nntxpk

trnktxrktxpktxptCk

ttxp

n

nna

nb

nn

)(1

)()(),()()( 14

19

19

123 trk

nntrktpktxptCk

ttr nn

an

bn

n++++

+−−+=

∂∂

tkROOHROOH CtC 1exp)0()( −=


175 December 2015

(4. 26) (4. 27)

Assuming that the initiation is much faster than the hydrogen abstraction, i.e., k1>> k3, then equations (4.26) and (4.27) can be rearranged as: (4. 28)

Solving (4.28) leads to: (4. 29) Considering quasi-stationary state, the macroradical concentration with time does not change significantly, then: (4. 30) Replacing in equation (4.24) and solving for r1(t): (4. 31) and substituting equation (4.31) in equation (4.23) for zero moment (n=0) :

'),'()()()()(

021 dxtxpxktCtCk

ttC

ROROOHRO

∞

•

• −=∂

∂

'),'()()()()(

031 dxtxpxktCtCk

ttC

HOROOHHO

∞

•

• −=∂

∂

tkROOHROOH

HORO CktCkt

tCt

tC1exp)0()(

)()(11

−==∂

∂=

∂∂ ••

( ) ( )tktkROOHHORO CCtCtCtC 11 exp1exp1)0()()()( 0

−− −=−=== ••

0)()()( 0

=∂

∂=∂

∂=∂

∂ttr

ttr

ttrn

)()()( 1

9

9231 tpk

ktCktra

b

+=


176 December 2015

(4. 32)Now, because the initial conditions are known, and assuming that the first moment is invariable, i.e., the total mass does not change with time, equation (4.32) can be solved by integration in time as:

(4. 33)

(4. 34)Assuming no thermal degradation, k0 might be neglected then equation (4.34) can be expressed in terms of Ns as:

(4. 35) Denoting Ka=k4k23/k9a and Kb=k4k9b/k9a, the following relation is obtained:

(4. 36)

( )09239

410

)()(),( kktCkk

ktpt

txpb

a

++=∂

∂

( ) ∂++=∂t

ba

t

tkktCkk

ktptxp0

09239

41

0

0 )()(),(

( )

+−++==−

−

1

023

1

02309023

9

1400

1)()0()(k

eCkkCktkkCk

ktpktptp

tk

ba

19

0234

19

0234

9

94

9

0234

0

11

)(1)(

kkCkke

kkCkkt

kkk

kCkk

MtMtNs

a

tk

aa

b

ann

−+

+=−= −

( )1

0

1

00

1)(kCKe

kCKtKCKtNs atka

ba −++= −


177 December 2015

Another simplification that can be done is assuming that hydrogen abstraction from the primary radicals is significantly faster than macroradical chain transfer, that is, k23>> k9b or Ka>>Kb, then the (4. 35 is simplified to:

(4. 37)Considering that for the same temperature, all kinetic constants should be the same, then, changing the concentration of hydroperoxide allows calculating the Ka constant.

(4. 38)

Figure 4. 10. Ka calculation for SBR degradation at 100 ºC and different concentrations of hydroperoxide from equation (4.38), the dashed line is the fitting until 15000 seconds. The solid line represents the fit until 23000 seconds.

19

0234

19

0234

9

0234 1)(kkCkke

kkCkkt

kCkktNs

a

tk

aa

−+= −

−+= −

110

11)(1

ke

ktK

CtNs tk

a

0 5000 10000 15000 20000 25000

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Ns/

c

time(s)

−+= −

110

11)(1

ke

ktK

CtNs tk

a


178 December 2015

Figure 4. 11. Ka and Kb calculation for SBR degradation at 100ºC and different concentrations of hydroperoxide from equation 4.36 Figure 4.10 shows the results for this approximation. It can be seen that deviation of the theoretical behavior occurs after 15000 seconds. For the dashed line, the value of Ka is 6.65x10-4 with a R2 of 0.90616, while the solid line, fitted the curve for longer times and leads to a Ka value of 3.88x10-4 with a R2 of 0.83235. Performing the same approximation but using Equation (4.46) directly, it is possible to calculate Ka and Kb at the same time, as shown in Figure 4.11. Results from this approximation can be detailed in Table 2. Table 2. Curve fitting for kinetic constants calculation Ka Kb R2 ª100ºC until 25000 s 3.88x10-4 0 0.83235 ª100ºC until 15000 s 6.65x10-4 0 0.9062 b100ºC 0.28% (4. 36 1.67x10-4 1.23x10-7 0.9906 b100ºC 0.32% 6.32x10-6 1.09x10-9 0.9804 b100ºC 0.50% 1.06x10-6 4.06x10-15 0.9334 b120ºC 0.20 % 1.16x10-5 1.05x10-9 0.7427 b120ºC 0.60% 7.65x10-5 1.17x10-9 0.8735 b100 ºC 10% Alkylbenzene 6.67x10-25 7.10x10-11 0.9617 b100 ºC 20% Alkylbenzene 7.10x10-6 1.00x10-22 0.9273 ª Calculated from equation 4.38 b Calculated from equation 4.36

0 5000 10000 15000 20000 25000

0.0

1.0x10-5

2.0x10-5

3.0x10-5

0.28% HPC 0.32% HPC 0.50% HPC

Ns

time(s)

( )1

0

1

00

1)(kCKe

kCKtKCKtNs atka

ba −++= −


179 December 2015

At first glance, it can be confirmed that the order of magnitude of Ka is higher that Kb, in accordance to the approximation given by equation (4.38). On the other hand, the difference observed in those fittings may arise from the rather large experimental errors involved, as shown in Figure 4.3. For higher temperatures, i.e., 120 ºC, the kinetic constant for thermal degradation k0 should not be neglected, thus it should be counted as included in the Kb constant. A higher deviation from the theoretical approximation was observed for 120 ºC, as can be seen in the R2 values of the curve fittings. Figure 4.12 shows the curve fittings for the SBR degradation at 120 ºC.

Figure 4.12. PBE approximation for SBR degradation at 120ºC. Finally, equation (4.36) was used to theoretically calculate the retardation of the reaction with addition of alkylbenzene as transfer agent. The aromatic group can stabilize a secondary radical in the α carbon of the aromatic ring through electron induction effect, involving another hydrogen abstraction step and additional intermediate radical species to the general reaction. Radicals in solvent molecules are more stable than on the polymer chain, and this effect retards the reactions of

0 10000 20000 30000 40000

0.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

5.0x10-5

6.0x10-5

Ns

time (s)

0 % 0.2 % HPC 0.6 % HPC 0% (fit Equation 23) 0.2% (fit Equation 40) 0.6% (fit Equation 40)


180 December 2015

equations (4.10), (4.11) and (4.12) shifting both k23 and k4 to lower values. Similar results have been obtained for polystyrene degradation(Sato, Murakata, Baba 1990) where hydrogen donor solvents decreased the polymer degradation. For Figure 13 shows the fitting from equation (4.36) and the results are summarized in Table 2. Despite the fact that no clear trend can be drawn from the kinetic constant, it is important to point out the magnitude of the shift. The larger shifting effect occurs for Kb, which means that the reaction of equation (4.12) is the most affected by the addition of the transfer agent.

Figure 4. 13. Curve fitting for the SBR degradation with 0.50% of hydroperoxide at 100 ºC and different concentrations of alkylbenzene in the solvent. From this set of experiments, the experimental error or the raw polymer heterogeneities were too high; then, the calculation of the kinetic constant has been compromised. Nevertheless, some results from this experimental work can be highlighted as follows: • No thermal degradation was observed below 100ºC, but for 120 ºC, the effect of temperature starts to be significant. • Addition of an initiator like cumene hydroperoxide leads to a significant increase in polymer degradation.

-5000 0 5000 10000 15000 20000 25000 30000 35000-2.0x10-6

0.0

2.0x10-6

4.0x10-6

6.0x10-6

8.0x10-6

1.0x10-5

1.2x10-5

1.4x10-5

1.6x10-5

Pure alyphatic solvent 10% Alkylbenzene 20% Alkylbenzene

Ns

Time (s)


181 December 2015

• In the concentration range of hydroperoxide studied, no significant differences were found for the induced degradation • Addition of a transfer agent, such as an akylbenzene in the solvent, retards significantly the degradation process.

Conclusions

In this work, the characterization of the degradation reaction of SBR in solution was studied by the changes in molecular weight and its distribution, despite the fact that chemical changes were not detectable by spectroscopic techniques. The population balance equation analysis is an important tool to quantify the degradation process from the changes in molecular weight distribution. Despite large experimental errors, it can be concluded that the degradation of SBR in solution in the absence of oxygen occurs mainly by random chain scission events which are dependent on the molecular weight. The primary radicals from the hydroperoxide decomposition lead to the formation of macro-radicals in the main chain, which may generate new smaller chains from their scission. The results confirm the hypothesis of the degradation mechanism expected for SBR. A simple transfer agent as an aromatic solvent can be used during specific applications to retard SBR degradation, either to retard the SBR degradation at high temperatures or to control the degradation kinetics. Acknowledgements

The authors would like to thank to PDVSA Intevep, for funding through the ID-6085-0000 project in the Department of Well Productivity. Also FONACIT (Science and Technology Ministry of the Bolivarian Republic of Venezuela) and Ministry of Foreign Affairs (France) for funding received through project: PCP 2011001409 (Postgraduate Cooperation Project).


182 December 2015

References

CHIANTORE, O, et al. Mechanism and molecular weight model for thermal oxidation of linear ethylene-butene copolymer. Polymer. 2001. vol. 42, p. 3981–3987. CHODAK, I and ZIMANYOVA, E, The effect of temperature on peroxide initiated crosslinking of polypropylene. European Polymer Journal. vol. 20, p. 81–84. GORDON C. G; MEYER, J.M and MCWALTER, I.T. Thermal degradation of polystyrene. 3. A Reappraisal. Macromolecules. 1978. pp 696–700. JIANG, D.D. et al. Thermal decomposition of cross-linked polybutadiene and its copolymers. Polymer Degradation and Stability . 1999. vol. 65, pp. 387–394. KARMORE, Vishal; MADRAS, Giridhar. Kinetics for thermal degradation of polystyrene in presence of p-toluene sulfonic acid. Advances in Environmental Research, 2002, vol. 7, no 1, pp. 117-121. KARMORE, Vishal and MADRAS, Giridhar, Thermal Degradation of Polystyrene by Lewis Acids in Solution. Industrial & Engineering Chemistry Research. 2002. Vol. 41, , pp. 657–660. KIM, Young-Chul and MCCOY, Benjamin J. Degradation Kinetics Enhancement of Polystyrene by Peroxide Addition. Industrial & Engineering Chemistry Research 2000. vol. 39, pp. 2811–2816. KODERA, Yoichi and MCCOY, Benjamin J. Distribution kinetics of radical mechanisms: Reversible polymer decomposition. AIChE Journal 1997. vol. 43, pp. 3205–3214. MADRAS, Giridhar and CHATTOPADHYAY, Sujay, Optimum temperature for oxidative degradation of poly(vinyl acetate) in solution. Chemical Engineering Science. 2001. vol. 56, pp. 5085–5089. MADRAS, Giridhar,et al. Molecular Weight Effect on the Dynamics of Polysterene Degradation. Industrial Engineering Chemistry Research. 1997. vol. 36, pp. 2019–2024. MADRAS, Giridhar and MCCOY, Benjamin J., 1997, Oxidative degradation kinetics of polystyrene in solution. Chemical Engineering Science. 1997. vol. 52,p p. 2707–2713. MADRAS, Giridhar and MCCOY, Benjamin J. Time evolution to similarity solutions for polymer degradation. AIChE Journal. 1998. vol. 44,pp. 647–655. MADRAS, Giridhar and MCCOY, Benjamin J. Kinetics and dynamics of gelation reactions. Chemical Engineering Science 2007. vol. 62, pp. 5257–5263.


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MADRAS, Giridhar, SMITH, J.M. and MCCOY, Benjamin J. Thermal degradation kinetics of polystyrene in solution. Polymer Degradation and Stability. 1997. vol. 58, pp. 131–138. MADRAS, Giridhar, SMITH, JM and MCCOY, BJ., Thermal degradation of poly (a-methylstyrene) in solution. Polymer degradation and stability [online]. 1996. vol. 52.. MCCOY, Benjamin J. and MADRAS, Giridhar. Degradation kinetics of polymers in solution: Dynamics of molecular weight distributions. AIChE Journal. 1997. vol. 43, pp. 802–810. MCCOY, BJ, Continuous kinetics of cracking reactions: Thermolysis and pyrolysis. Chemical engineering science . 1996. vol. 5, p p. 2903–2908. MCCOY, BENJAMIN J. WANG, Ming, Continuous-Mixture fragmentation kinetics: particle size reduction and molecular cracking. Chemical engineering science. 1994. vol. 49, pp. 3773–3785. RAMKRISHNA, Doraiswami, 2000, Population balances: Theory and applications to particulate systems in engineering. San Diego, CA. USA : Academic Press. SATO, Shimio, et al. Solvent effect on thermal degradation of polystyrene.Journal of applied polymer science, 1990, vol. 40, no 11-12, p. 2065-2071. SCHNABEL, Wolfram, 1981, Polymer Degradation, Principles and Practical Applications. New York. SMAGALA, Thomas G.; MCCOY, Benjamin J. Mechanisms and approximations in macromolecular reactions: reversible initiation, chain scission, and hydrogen abstraction. Industrial & engineering chemistry research, 2003, vol. 42, pp. 2461-2469 STERLING, W. Jerome and MCCOY, Benjamin J. Distribution kinetics of thermolytic macromolecular reactions. AIChE Journal 2001. Vol. 47, pp. 2289–2303. TOBITA, Hidetaka, Simulation model for the modification of polymers via crosslinking and degradation. Polymer.1995. vol. 36, p p. 2585–2596.

184 December 2015

CHAPTER V

Oil-based self-degradable gels as

diverting agents for oil well operations

Oscar Vernáez, Augusto García, Franklin Castillo, María

L.Ventresca, A.J Müller

Doctoral student contribution: In this chapter, the student designed and performed 90% of the experimental section, along with the analysis of all the results. He also wrote the paper as corresponding author. The paper has been submitted to the Journal of Petroleum Science and Engineering.

Chapter V: Oil-based self-degradable gels as diverting agents for oil well operations

185 December 2015

ABSTRACT

Different formulations of oil-based self-degradable gels have been evaluated as possible diverting agents during oil wells operations. The time dependent rheological behavior of the gels was measured to verify polymer crosslinking, maximum gel strength and gel degradation, all of which can be adjusted by varying formulation depending on operation needs. Single core tests were performed to evaluate pressure resistance as a function of gel strength for different saturation conditions. Parallel cores tests were also carried out to validate diversion efficiency and mobility restoration. Gel degradation in porous media was evaluated for single core tests and a simple scaling factor was applied to calculate productivity recovery during degradation. Compatibility tests with acids and corrosion tests were also performed. Additionally, a remedial formulation to accelerate gel degradation was evaluated. The pressure resistance of the protected zone was determined by gel strength, invasion depth and initial saturation, and it may be scaled linearly. Both diversion and productivity recovery can be considered successful. The remedial formulation accelerates the gel degradation process, but it should be injected at high pressure conditions. From the overall results, it is possible to conclude that the formulations tested in this work are suitable for pilot tests in the field.


186 December 2015

RESUMEN

Se evaluaron diferentes formulaciones de un gel autodegradable base aceite como posible agente de divergencia a ser utilizado en operaciones en pozos petroleros. Se midió el comportamiento reológico de los geles en función del tiempo para observar el entrecruzamiento del sistema polimérico, la máxima consistencia del gel y su degradación, parámetros que pueden ser ajustados mediante la variación de la formulación en función de las necesidades operacionales. Se realizaron evaluaciones de medio poroso en un solo núcleo para evaluar la resistencia a la presión como función de la consistencia del gel para diferentes condiciones de saturación. También se realizaron ensayos en núcleos paralelos para validar la eficiencia en la divergencia y la restauración de la movilidad. La degradación del gel en el medio porosos fue evaluada en ensayos de un solo núcleo y se calculó la restauración de la productividad durante la degradación mediante un factor de escalamiento simple. Adicionalmente, se evaluó una formulación de remediación para acelerar la degradación del gel. La resistencia a la presión de una zona protegida con este gel puede ser escalada linealmente y se determina a partir de la consistencia del gel, el radio de invasión y la saturación inicial. Tanto la divergencia como la restauración de la productividad se consideraron exitosas. La formulación de remediación acelera el proceso de degradación del gel, pero debe ser inyectada a altas presiones. A partir de los resultados globales, es posible concluir que las formulaciones evaluadas en este trabajo pueden ser utilizadas en pruebas piloto de campo.


187 December 2015

RÉSUMÉ

Différentes formulations de gels auto-dégradables à base d’huile ont été évaluées en tant que possible agents de dérivation pour des opérations de traitement de puits. Le comportement rhéologique des gels a été mesuré en fonction du temps, pour vérifier la réticulation du polymère, la force de gel maximale, et la dégradation du gel ; toutes ces valeurs peuvent être ajustées aux nécessités de l’opération de puits en jouant sur la formulation. Des test d’écoulement dans une carotte simples ont été réalisés pour évaluer la résistance à la pression en fonction de la force de gel, dans différentes conditions de saturation. Des tests d’écoulement dans carottes parallèles ont également été menés, afin de valider l’efficacité de la dérivation et la restauration de la mobilité. La dégradation du gel en milieu poreux a été évaluée pour les tests des carotte simples, et un simple facteur d’échelle a été appliqué pour calculé la récupération de productivité au cours de la dégradation. Des tests de compatibilité avec des acides, ainsi que des tests de corrosion a également en mis en œuvre. De plus, une formulation alternative permettant l’accélération de la dégradation du gel a été étudiée. La résistance à la pression de la zone protégée a été déterminée à partir de la force du gel, de la profondeur de pénétration et de la saturation initiale, et peut être décrite par une relation linéaire. La dérivation et la récupération de productivité peuvent être considérées comme des succès. La formulation alternative accélère le processus de dégradation du gel, mais il faudrait l’injecter à haute pression. A partir de l’ensemble de ces résultats, nous pouvons conclure que les formulations testées dans ces travaux sont aptes aux tests pilotes sur champ.


188 December 2015

Introduction

Increasing oil well productivity can be achieved by producing more than one interval in the same completion. If the intervals have different petrophysic characteristics like different pressure, permeability, formation damage or water invasion, well intervention will be necessary to treat a particular zone within the completion. Depending on the type of treatment and completion configuration, it is sometimes mandatory to isolate a specific productive interval to divert a chemical treatment to the desired zone. This could be done in order to protect the productive zone from the chemical treatment or to maximize the placement of the treatment in the desired interval. In the oil industry, several techniques are used to divert fluids to specific intervals in the oil well. Kalfayan and Martin (2009) have reviewed most of the methods found in literature. Mechanical methods as gravel packers, ball sealers and inflatable packers have many advantages but require specific completion features to allow their use, and they can also be expensive or not accurate. Allison et al. (2011) published an overview chart with new criteria for selecting chemical diversion methods. Chemical diversion methods, including benzoic acid flakes, foams, viscoelastic surfactants, self viscosifying acids and viscous pills, have been widely used in industry. Viscoelastic surfactants and self diverting acids are designed for acid stimulation treatments only (Liu et al. 2013; Hull et al. 2015), leaving aside other applications such as water shutoff or non-reactive stimulation. Other methods, like polymer viscous pills, require injection of a breaker to return productivity to the isolated zone, leading to inhomogeneous removal and formation damage. More recently, relative permeability modifiers have been also used to divert acid stimulations in operations called conformance while stimulating (Cuadras et al. 2015)


189 December 2015

Diversion can be useful in a variety of treatments, like stimulation, water shutoff treatments, and fluid loss in low pressure zones during drilling or cementation, among others. Diversion methods should avoid the inflow of a chemical treatment in the isolated zone and support the bottom-hole pressure induced during injection. It is also important to return the isolated interval to its original productivity. The deferred production time of the protected zone could be a major economic problem. A typical concern in operations can arise in complex treatments, where diversion time might be undefined or too long for a regular application. Thus, it is important to prepare formulations that can be placed in a specific interval to protect it for an undefined time. In such cases, after the desired protection time, a remedial formulation can be injected to remove the diversion treatment. An example could be water shutoff treatments in complex completions, where a productive zone needs to be protected for a prolonged period of time. If the protected zone returns to its original permeability before the water shutoff treatment is done, some of this treatment can flow in the productive zone creating damage or even blocking it completely. In this work, different formulations of an oil-based self-reversible gel are tested in porous media flow to evaluate their behavior as diverting agents for oil well chemical treatments. Rheological behavior and compatibility with different fluids are also evaluated. The gels are obtained by covalently crosslinking an organic polymer in solution that can later be degraded by addition of a degradation agent into the formulation (Ventresca et al. 2009) Crosslinking time, maximum strength and degradation time are adjusted by controlling the crosslinking and degradation reaction kinetics at a specific temperature. Changing concentration of both initiator and degradation agents, generates a competition between crosslinking and degradation reactions which allows control of the rheological behavior of the gelling system (Vernáez et al. 2015a).


190 December 2015

The observation of a maximum in complex modulus is related to the maximum pressure supported by the diversion gel placed in porous media. Gel degradation is evaluated in porous media by measuring the permeability return to its original value (i.e., before the gel is formed). For those cases when isolation needs to be kept for an undefined period of time, a non-reversible gel was also evaluated and degradation was later accomplished by injecting and soaking the porous media with a remedial formulation. Experimental

Materials The diverting gels were formulated to obtain different maximum strengths at specific temperatures. Therefore, the polymer concentration, initiator concentration and degradation agent were selected to fit specific requirements. Cold emulsion polymerized styrene-butadiene rubber (SBR-8113) from ISP was used. Peroxides used as crosslinking agents were dicumyl peroxide, di-tert-butyl peroxide and di-lauryl peroxide while cumene hydroperoxide was used as degradation agent, all from Akzo Nobel. Details on their preparation have been published recently (Vernáez et al. 2015a)The remedial formulation contains cumene hydroperoxide dissolved in inert solvents. The viscosity of the polymer solutions vary from 40 to 65 cP at 25ºC depending on polymer concentration. Dehydrated dead oil with 26ºAPI and 38 cP at 25 ºC was used for some core tests while regular diesel (3.3 cP at 25 ºC and 1 cP at 95ºC) was used for others. An oil-based drilling mud was prepared from a water-in-oil emulsion with brine, organophillic clay, emulsifiers and baryte, among others components. A non-reactive stimulating agent, i.e., Ultramix®, was used to remove formation damage.

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192 December 2015

A core plug permeameter was used to measure pressure drops through porous media. The permeameter was modified to measure both single core and parallel cores and it is shown schematically in Figure 5.1. The initial state for all tests implies brine injection followed by oil injection until irreducible water saturation (Siw) is reached. The flow rate used for the initial state was 0.5 mL/min and was injected in an arbitrarily “production direction”. The inflow and outflow lines of both core holders are connected in such way that inlet and outlet pressures would be the same for both. Outflow rates were obtained by measuring outlet volume as a function of time. A nitrogen bottle was used to set the back-pressure to 200 psi. Methods

Single core experiments After the initial state is achieved, one porous volume (PV) of the diversion gel formulation was injected to the core in opposite direction to the previous fluid injection, arbitrarily named “injection direction”. The formulation is then allowed to crosslink with temperature. After reaction, diesel or oil is injected to the core in the injection direction and the pressure drop throughout the core is measured. The same procedure but in the production direction was repeated during several days to evaluate gel degradation or productivity recovery. Different formulations were evaluated, in which the maximum gel strength varied for each formulation. In another test, the initial saturation was evaluated by injecting water after the initial state in the production direction until residual oil saturation (Sor) was reached. Then, one PV of diversion gel formulation was injected to the core, keeping the rest of the test identical. For this test, the same formulation was used for both saturation conditions.


193 December 2015

Parallel cores experiments In the first parallel core test, two cores with different permeabilities (216 mD and 49 mD) were placed in core holders in a procedure similar to that presented by Siddiqui et al. (2003) or by Alvarez et al. (2000). During the initial state, an inflow rate of 0.5 mL/min was injected to the system, while pressure drops and outflow rates in each core were measured. One porous volume was injected to the system at 0.5 mL/min in the injection direction with both cores open. After gelation time, oil was injected in the production direction for several hours, while pressure drops and outflow rates were evaluated for each core. Another parallel cores test was performed to simulate a typical stimulation treatment. In this test, two cores of similar permeability (around 200 mD) were placed in core holders. After the initial state was reached, a drilling mud was forced through core number 2 only, to induce formation damage. Then, one PV of diversion gel formulation was injected to the system with both cores open. After gelation time, four cycles of 2 PV of the stimulation agent were injected in the injection direction every 2 hours. After 12 hours since the last stimulation, diesel was injected in the production direction to measure the final state for the next 50 to 72 hours. Pressure drops and outflow rates for each core were evaluated during the test. All parallel core experiments were conducted at 149 ºC and the same diversion gel formulation was used. Remedial application An additional single core test was performed to evaluate the degradation ability of a remedial formulation applied to an in-situ diversion gel. 6 PV of the formulation with no degradation agent was injected to the core at 4 mL/min after the initial state. The temperature was kept at 138 ºC. The rheological behavior of this formulation was evaluated with Sydansk (1990) qualitative scale, with a maximum strength equivalent to a “slightly deformable non-flowing gel”.


194 December 2015

The pressure test was performed after three days. Then, 0.5 PV of a remedial solution based on the degradation agent was injected into the core and left to soak for 2 hours followed by a pressure test. This remedial procedure was repeated 3 more times for the next 6 days and productivity recovery was calculated after each soak. Compatibility test Diversion gel formulations were prepared in contact with a mixture of hydrochloric acid and hydrofluoric acid (1:1) to see if their rheological behavior was affected by the presence of acid. A low temperature formulation and the acid mixture were added to a flask and sealed under nitrogen atmosphere and placed in an oven at 90ºC. Qualitative observations of gel strength during crosslinking and degradation reactions were reported. Corrosion tests were also performed. Samples of metal from typical oil well completion pipes were soaked in different formulations of diversion gels, in degradation agent only and in remedial formulation, all in an oxygen free atmosphere at 90 ºC. Visual inspection of metal samples was reported. The rheological behavior of diversion gels was evaluated mixing them with different diesel fractions to evaluate the dilution effect. Four mixtures of diversion gel formulation and diesel were evaluated with diversion gel/diesel ratios of 2:1, 3:1, 4:1 and 5:1. Each mixture was added into different flasks, sealed under nitrogen atmosphere and placed in an oven at 149ºC. Samples were taken at different times and were evaluated with dynamic oscillatory rheology tests in their linear viscoelastic range.


195 December 2015

Results and discussion

Chemical crosslink and degradation reactions The polymer solution (i.e., the gel forming formulation) with all its additives is designed to have a low viscosity during injection in the wellbore. The viscosity will change as a function of polymer concentration. Additives do not contribute to the solution viscosity before crosslinking. Figure 5.2 shows an example of a formulation with 2.5% of polymer at different concentrations. The inner figure presents an Arrhenius type fit, which can be used to calculate the viscosity at other temperatures. Viscosity is an important parameter for injectivity because down-hole pressure will depend linearly on it. Depending on the application temperature and application design, the kinetics of crosslink and degradation reactions are adjusted by regulating the concentration of crosslinking, degradation and transfer agent. Both crosslinking and degradation reactions are temperature induced radical reactions, in which peroxides decomposes with temperature to create radicals. These radicals abstract hydrogens from the polymer chains to create macroradicals. In optimal concentrations, these macroradicals can combine with one another to form covalent bonds between polymer chains (i.e., a termination reaction). After several termination reactions, a three dimensional network is formed, and the polymer structure becomes a solid. Mobility of polymer chains is then constraint and termination reactions become unlikely, leading to β-scission reactions. The hydroperoxide decomposes at longer times to create new radicals which promote β-scissions. The polymer network starts to degrade itself, forming low molecular weight products, which are again soluble or dispersible in an organic media (Vernáez et al. 2015b).


196 December 2015

Figure 5. 2. Viscosity behavior for a formulation with 2.50 wt. % of polymer at different temperatures The initial gelation time is the time needed to reach the liquid-solid transition at a specific temperature. At this point, viscosity diverts to an infinite value and the formulation is no longer liquid. Chemically, initial gelation time can be described as the time where all polymer molecules in solution have reacted with each other to create a unique three dimensional network with solvent in its interior. In terms of rheological characterization, at this point, the elastic or storage component of the shear modulus become higher than the viscous component at a specific shear rate and deformation comparable to those imposed to the fluid during the injection or once it is placed in the porous media. Then, it is important to set this initial gelation time to be long enough to allow the injection and placement of the fluid in the desired zone. The time needed to reach the maximum strength of the polymer network is denoted maximum gelation time and it depends on temperature and additive concentrations. Gel strength can be expressed in terms of complex modulus. It defines the deformation capability of the polymer network, and determines the pressure resistance during the diversion application. The higher the gel strength, the higher the pressure it can resist before allowing any fluid to flow through the porous media. However, high gel strength could also lead to longer degradation times.

1 10 10020

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0.0030 0.0032 0.00341.4

1.5

1.6

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1.8

Log

η

1/T (1/K)

η=0.0732exp(872.6/T)R2=0.9999

Visc

osity

(cP)

Shear rate (1/s)

25 ºC 50 ºC 70 ºC


197 December 2015

Figure 5. 3. Left: Rheological behavior of a gel formulation at 10 rad/s. Right: Frequency sweep obtained by small angle oscillatory rheometry The time the polymer network takes to degrade is defined as degradation time. As the gel degrades, it forms low molecular weight oil-soluble products, which can flow again under a specific flow rate. It depends on the maximum strength of the polymer network and also depends on the flow stress or pressure imposed over it. Figure 5.3 (left) shows the rheological behavior of a typical formulation, three important times can be highlighted in the curve. The initial gelation time, represented by the first crossover between storage modulus G’ and loss modulus G’’. From that point onwards, the polymer system may be consider as a gel. At longer times, the highest strength is achieved, represented by a maximum in G’. Finally, degradation sets in and G’ decreases until a second crossover between G’ and G’’ defines the time at which degradation is complete and the sample is no longer a gel, but recovers its liquid state. Figure 5.3 (right) presents the frequency sweep characterization in oscillatory shear tests for three different times. Both the initial (2 h) and final behavior (36h) are almost identical, since they represent the typical behavior of viscous polymer solutions. At intermediate times (6h), this particular formulation behaves as a gel as denoted by the frequency independent behavior of G’ (in addition to the fact that G’ is much higher than G’’).

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G'>G'' gel

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Gel point G'=G'' Degradation

Polymer in solution

1 10 1000.1

1

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G' G'' 02 hours 06 hours 36 hours

G',

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Pa)

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198 December 2015

Single core experiments Formulations of diversion gels with different rheological behavior were injected to core samples with similar petrophysical properties. The injected volume was equal to one porous volume (PV) of each core. The polymer solution was injected at irreducible water saturation (Siw), and no significant change in saturation was observed during polymer injection because of its low viscosity. It is assumed that the polymer solution will remain in the oil saturated zones inside the core, due to its organic affinity, as predicted by relative permeability and segregated path (Lian et al. 1994). During the injection of the polymer solution, a part of it was observed in the outlet before an injection volume equal to PV-(PV.Siw) was injected, which means that the formulation did not reach all oil saturated pores in the core, i.e., an inaccessible porous volume exists. Once the formulation is located inside the porous media, temperature activates the crosslinking reaction until the gel reaches its maximum strength. The polymer gel mechanically blocks the porous space, decreasing the effective area for flow and therefore reducing permeability to almost zero. Figure 5.4 shows the behavior of the pressure drop for gels with different strengths. The pressure drop has been normalized to psi/ft. The flow resistance increases proportionally to the gel strength expressed in terms of complex modulus values. Two samples show a maximum pressure from which fluids start to flow at the outlet of the core. This pressure threshold behaves similar to the yield strength of materials. The outflow rate eventually reaches a steady state. For the sample with 8.4 Pa of gel strength, only a change in the slope can be observed at a critical pressure. In this case, the steady state was not reached. For the lowest gel strength, no pressure threshold was observed and the system quickly reaches the steady state.


199 December 2015

Figure 5. 4. Normalized pressure drops for different maximum gel strengths expressed as complex modulus at a frequency of 10 rad/s These behaviors are related not only to the maximum gel strength, but also to the gel elongation. Gels located in porous throats are deformed by the pressure; once the maximum strain is reached, flow starts and pressure should be maintained to keep this deformation. Gel deformation at porous throats increases the effective flow area, creating a contact fluid path. If pressure is removed, the outflow rate instantly stops. Zitha et al. (2002) described gel flow through a porous medium by three mechanisms: elastic compressive deformation of the medium, microscopic flow through the polymer network and macroscopic displacement of gel by the fluid. Gel displacement is not reversible and the flow of solvent through the gel is assumed negligible. Therefore, it is possible to assume that gel flow is only due to elastic compressive deformation of the medium. The pressure threshold can be characterized as a critical yield pressure, treating the interconnecting gel saturation as a deformable material with a G* modulus, or its equivalent compressibility coefficient. Thus, critical differential pressure is

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200 December 2015

proportional to the volume occupied by the gel. Linear normalization will then be valid for cylindrical linear core plugs, and it will underestimate the pressure resistance in a radial system, especially near-wellbore formation, where porous volume increases quadratically with invasion depth. For example if a gel filled core plug can resist a differential pressure of 3000 psi/ft, it is expected that 1 ft of radial invasion will resist at least 3000 psi/ft. Effect of initial core saturation Gel location within the porous medium is an important factor determining diversion efficiency, because the flow will depend on the volumetric fraction blocked by the diversion gel. As stated before, a critical volumetric deformation of the gel should be achieved to expand the effective flow area. In Figure 5.5, two core tests are compared, with identical diversion gel formulations, but injected at different saturation conditions. When the diversion gel is injected under irreducible water saturation, it covers more porous volume reducing effective flow area, while injection in residual oil saturation comprises less porous volume and effective flow area is higher, less deformation energy is necessary to increase effective flow area and critical pressure is then lower. It can be also assumed that if more than one porous volume of the diversion gel is injected to the core, water saturation will be progressively displaced to lower values until it eventually reaches the same conditions. This result implies that injection of this organic-based diversion gel will self-divert to preferentially invade the oil-saturated matrix. From these results, it is recommended to inject a pre-flux of diesel to modify saturation conditions near wellbore to enhanced diversion.


201 December 2015

Figure 5. 5. Pressure test for two saturation conditions during diversion gel injection. Maximum gel strength G*=7.5 Pa at 10 rad/s and 90 ºC. Two parallel cores test and stimulation simulation For the first parallel core test presented in Figure 5.6, two plugs with different permeabilities were used. During the initial state, the pressure drop will be the same for both plugs and the flow rates are determined by Darcy’s law. However, despite that the permeability ratio is about 5, the mobility ratio is 2.5, indicating a different effective permeability ratio, or different saturation conditions. Hence, the diversion gel formulation injected is fractionated in 0.64 to the high permeability core and 0.36 to the low permeability one. After crosslinking, diversion is achieved; reversing the flow ratio and directing 95 % of the fluid to the originally low permeability core. As gel degrades, mobility in the high permeability core returns to its original value, while mobility in the low permeability core is reduced by 42%. Mobility restoration in the high permeability core is an evidence of gel degradation but the gel fraction located in the low permeability core creates a damage which did not recover completely during observation time. It is important to take into account the permeability for future works, because it could determine the restoration time. During gel degradation, it is possible to observe expulsion of the gel by the injected fluid. Such gel expulsion was

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202 December 2015

proportional to core permeability, as in the mechanism described by Zitha et al.(2002). Thus, even if the gel degradation state is the same in both cores, only in the high permeability core the degraded gel is expulsed by fluid, while in the low permeability core, the gel should degrade more to be able to flow out of the porous media.

Figure 5. 6. Parallel cores test with two permeabilities. Core 1, high permeability K=216 mD; Core 2 Low permeability K=49 mD For the next parallel core test, two plugs with similar petrophysical properties were used. At the initial state, inlet fluid is allowed to flow through the system, and flow rate is measured in each core. Because the pressure drop is the same, the flow rate is determined by Darcy’s law within each porous medium. When formation damage is induced in one core, a skin factor is created and the permeability is significantly reduced. Fluids will then flow preferentially through the undamaged core. When, one porous volume of the diversion gel is injected to the system it flows almost completely through the undamaged core as it can be seen in Figure 5.7. After gelation time, a non-reactive organic stimulation treatment is injected to the system at the same total flow rate. High differential pressure was needed to inject the stimulation fluid as indicated by the mobility reduction. The diversion gel formulation was selected to start degradation after 12 hours. Then diesel is injected during the next two days to evaluate the mobility recovery in the cores. Mobility recovery did not reach the original values, because diesel should displace a more viscous polymer solution, which

0 10 20 30 40 50 600.050.100.150.200.250.300.350.400.450.50

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initial state

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203 December 2015

can be considered as formation damage. Nevertheless, from the practical point of view diversion, stimulation treatment and gel degradation were considered successful.

Figure 5. 7. Parallel cores test with stimulation. Formulation maximum gel strength (G* at 10 rad/s) of 9.8 Pa. Left: flow rates. Right: Mobility Field Application design Chemical diversion treatments have to be designed as a function of the petrophysical characteristics near wellbore, wellbore infrastructure and times needed to perform further treatments. For stimulation applications, bullheading injection can be used to temporally block high permeability zones, called thief zones, which may be prospective for oil production. In such cases, diversion gel placement will be naturally directed to thief zones because of their high permeability. The volume of the diversion gel is an important aspect to take into account. Too high volumes of diversion gel will invade and partially block zones which need to be stimulated and lower volumes may not be enough to block completely the thief zones. Petrophysical characterization is the key for choosing optimal volumes. Injection tests are also important to calculate the optimum diverting agent volume. These tests will relate the injection pressure to the injected volume. The fluid selection must be taken into account for the injection test. An injection test with water could lead to a

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204 December 2015

significantly different result than an injection test made with diesel or any organic fluid (even stimulation fluid if it is organic), because of the relative permeability conditions of the rock matrix. It is then important to perform the injection test with diesel or any organic fluid, to determine the optimal volume for this diversion gel. Gel strength has to be chosen depending on the permeability of the aimed zone, and on how much formation damage or skin factor is present. If the skin factor in the damaged zone is too high, the pressure needed to place the stimulation fluid in that zone should be accordingly high, and then the strength of the gel placed in the thief zone must be enough to support that pressure without letting the stimulation fluids get in the thief zone. The injected volume of the diversion agent can also determine the pressure that the blocked zone can resist. It is difficult to calculate the pressure transferred through a porous matrix filled with gel, because of the geometrical heterogeneities in porous media. As a rule of thumb, in this work the pressure resistance will be linearly extrapolated from core experiments and will be expressed as pressure by feet. Thus, each feet of gel placed in the formation will increase the pressure resistance of that zone. As it was mentioned before, pressure resistance is related to the effective flow area and deformability of polymer network; therefore, linear extrapolation as a practical rule will be valid and includes a security factor. If the skin factor of the damage zone is known, then the pressure needed to inject the stimulation fluid at a specific flow rate can also be determined. Then, the volume of diverting agent injected in the thief zone must be enough to support that pressure. Gel strength can be also increased, but it will affect the degradation time. Scaling factors and productivity recovery The effectiveness of diversion gels does not only depend on pressure resistance or how treatment fluids are directed to the desired zone, but also on how they degrade and flow out of the porous medium to restore the initial productivity of the zone


205 December 2015

where it was placed. Figures 5.6 and 5.7 show how the core with diversion gel restores the original flow rate. One way to evaluate gel degradation in cores is by comparing mobility or permeability at the initial and final state. Gel degrades progressively into low molecular weight products that dissolve or disperse in organic media. Reversing mobility or permeability is traduced in a productivity restoration of the zone where the diversion gel is placed. The kinetics of similar polymers degradation as those used here have been studied elsewhere (Vernáez et al. 2015b) and they are determined by temperature and additive concentration. From the chemical point of view, polymer degradation should always be effective as long as the degradation agent concentration remains in the polymer network. Experimentally, it has been proven that despite gel degradation can be accomplished in bottle reference experiments, still some damage can remain in porous media if viscous displacement is not effective. Nevertheless, it should be assumed that degraded gel will progressively flow out the porous medium. It is also assumed that for water-wet formations, no polymer adsorption on rock surface is expected. In the core, restoration can be evaluated as the quotient between permeability after gel placement and the original permeability (Ka /K0). In Figure 5.8 (above), restoration behavior of gels for single core experiments is shown. Even if only some of the tests restore more than half of the original permeability before 2 weeks, the trend of all gels is to degrade progressively with time. For parallel cores in Figures 5.6 and 5.7, mobility was restored almost completely for cores with diversion gel. During single core tests polymer solution is forced to enter in porous spaces which might be different from those reached by oil, because of its higher viscosity. During restoration test, oil cannot displace the degraded polymer solution due to the viscosity ratio, making difficult to remove it completely from porous spaces, even if the gel is totally dissolved. . For more viscous oil, the restoration is expected to be more effective. Diversion gel can be considered as a skin or formation damaged. During an application in oil wells, effectiveness of the treatment is associated to the invasion zone of the


206 December 2015

diversion gel. The zone treated with the diversion gel can be characterized by the productivity index J=Q/ΔP. Ideally, after gel degradation, productivity index should return to its original value, associated to a skin factor of zero. Skin factor s for radial matrix invasion can be calculated from the initial and final permeability and the invasion depth rd as:

−=

w

d

a rr

KKs ln10

(5.1)

where rw is the well radius. Oil well productivity index J can be expressed in terms of Darcy’s law for radial flow as a function of skin factor s, as:

+

=s

rr

hKJ

w

eln

20

μ

π

(5.2)

Where, h is the payzone length, μ is the viscosity and re is the drainage radius. Hence, productivity recovery can be calculated as the ratio Jb/Ja , where Jb is the productivity index before diversion gel injection and Ja is the productivity index after gel degradation.


207 December 2015

Figure 5. 8. Above: Permeability restoration for all single core tests. Below: Productivity recovery for all single core tests Assuming a skin factor s=0 before diversion gel application, the restoration of productivity of an interval with diversion gel can be calculated as a percentage of the original productivity index. Figure 5.8 (below) shows the productivity recovery for all single core tests presented in this work. It was assumed a hypothetical drainage radius of 1000 ft, a 7” production pipe and 1.5 ft of diversion gel radial invasion depth. It can be seen how productivity recovery of the zone treated with diversion gel is faster than permeability restoration. A restoration of 75% of the permeability in the core test will represent around 90% of productivity recovery in the well. It can also be pointed out that the invasion depth will not significantly affect productivity recovery. Numerical simulations and nodal analysis are recommended to calculate inflow curves from intervals, because drawdown may significantly change.

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208 December 2015

Because this type of treatment is designed for near well-bore application, another important scaling factor which needs to be taken into account is flow rate or linear velocity. For a specific drawdown, fluids flow from reservoir to the wellbore at a constant flow rate in steady state. Hence, linear velocity will increase quadratically as the fluid approaches the wellbore. The same phenomenon is observed for fluid injection. Experimentally, the flow rate for linear core plug tests has to be selected in accordance to flow rates expected during production and injection. Numerical simulators are helpful tools for estimating velocity during production and injection, especially in heterogeneous intervals. The estimation can also be done by assuming homogeneus radial invasion. A scaling factor for choosing appropriate flow rates in core tests can be done by considering linear velocity in the formation matrix. A reference invasion depth has to be selected to be the matrix element to be simulated in the permeameter. Knowing the length of the interval h, the well-bore diameter, the bottom-hole pressure and the reservoir pressure (or drawdown), and production or injection flow rate, it is possible to calculate the linear velocity of fluids ν at the reference invasion depth ri as follow: ( )wi rrhQ

+=

πν

2 (5.3) where Q is the production or injection flow rate. Then, the flow rate in a core test q can be calculated by normalization with the area A of the core plug.

Arq i )(ν= (5.4) Depending on the type of treatment to be made, the pressure resistance in the protected zone will be different. For stimulation treatment, formation damage will determine the pressure needed to inject the fluids in the formation. The upper limit for bottom-hole pressure is usually determined by the fracture pressure or maximum


209 December 2015

pumping capacity. If no information about damage is known, the diversion gel should be designed to resist a pressure near to the fracture pressure. The invasion depth can be then calculated from core pressure resistance tests. If fracture differential pressure is 1500 psi, the formulation should resist at least 1500 psi/ft. It is recommended to invade at least 1 ft in the formation with diversion gel, calculated as a radial matrix invasion or with numerical simulation tools. The strength of the diversion gel can be increased as well as invasion depth, but economically increasing strength is always cheaper than increasing invasion depth, and also minimizes the productivity recovery time. Remedial test The formulation chosen for remedial test had degradation times longer than a month. Also porous volumes injected in the core were maximized to ensure maximum saturation of the diversion gel in the core. The pressure test in Figure 5.9 (left) was performed at a higher rate to simulate a near wellbore condition after three days. It can be seen how the gel strength remains high after three days of crosslinking and the yield critical pressure was obtained at about 2400 psi/ft stabilizing around 2300 psi/ft. After injection of the remedial formulation, productivity recovery was evaluated after every soak. Figure 5.9 (right) shows how productivity is recovered in at least 60% in 6 days. The remedial treatment depends on soak and how it diffuses through the matrix of the diversion gel, therefore, high pressure should be imposed to the fluid to ensure penetration. It is still useful to have a remedial formulation to remove the diversion gel, in such cases where degradation could not be completely achieved or to reduce deferred production.


210 December 2015

Figure 5. 9. Left: Pressure test at a flow rate of 4 mL/min, after three days of injection. Formulation with no degradation agent and maximum strength equivalent to a “slightly deformable non-flowing gel. Right: Productivity recovery test taken after each soak with a degradation agent formulation Compatibility tests Because the diversion gel is mainly a network of styrene-butadiene rubber, no change in behavior was observed when it is exposed to acids. As crosslinking and degradation agents are oil soluble, both gelation time and maximum strength remained the same as the reference sample. Metal samples also remained unchanged after soaking with diversion gel formulation, which indicates no risk for metal components in the completion when using diversion gel. The sample in contact with pure cumene hydroperoxide was completely degraded. Products from homolitic decomposition of hydroperoxide react with the metal through the Fenton Redox reaction (De Laat and Gallard 1999). Iron oxidizes to Fe2+ and Fe3+, which react with the hydroperoxide in a continuous self catalyzed reaction leading to the total decomposition of the metal. When the metal sample was in contact with the remedial formulation, no evidence of oxidation was noticed, because it is formulated with inert solvents Dilution with diesel affected the behavior of the gel, because it reduced the local polymer concentration. Figure 5.10 shows the rheological behavior of a sample with different dilution fractions. The maximum gel strength is reduced significantly with dilution. It is important to take into account the dilution factor on the injection front,

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211 December 2015

as this will depend on the viscosity difference between the diversion gel formulation and the displaced oil in formation. Polymer diffusion can be negligible in porous media because of its high molecular weight, however peroxides diffusion can affect the behavior. Hence, it is recommended to displace some security volume of formulation, assuming some dilution or diffusion of the peroxides in the front, specifically in such formations with high permeability and porosity, where the dilution factor can be significant. It is meaningless to calculate it experimentally, because it will depend on fluids type, saturation and petrophysical properties of the matrix.

Figure 5. 10. Rheological behavior of a formulation at 149 ºC at different dilutions ratios with diesel Conclusions

Self-degradable gels have been formulated using an oil soluble polymer in solution along with initiator and degradation agents. The results obtained for core flood tests, are consistent with ex-situ rheological behavior, where maximum gel strength defines the pressure resistance of the diverting agent. Diversion and productivity recovery can be considered successful if scaling factors and other considerations as dilution

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(Pa)

Time (min)


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factors are taken into account. Some simple scaling rules have been presented which can be easily used during oilfield operations and for quick design of core test protocols using basic data. A remedial formulation to accelerate gel degradation can be considered as an alternative either to reduce deferred production times or to remediate residual damage caused by the diversion gel. Gel strength, invasion depth and initial saturation determined the pressure resistance of the protected zone, and may be scaled linearly, assuming volumetric deformation theory. From the overall results it was possible to postulate that formulations tested in this work are suitable for pilot tests in oil fields. Acknowledgment


ALLISON, David Blair, et al. Restimulation of wells using biodegradable particulates as temporary diverting agents. En Canadian Unconventional Resources Conference. Society of Petroleum Engineers, 2011. ALVAREZ, Jose M., et al. An optimal foam quality for diversion in matrix-acidizing projects. En SPE International Symposium on Formation Damage Control. Society of Petroleum Engineers, 2000. CUADRAS, Romel, et al. Selective Stimulation Using Self-Degrading Diverter Deployed with 1 1/2-in. CT During Completion Stage. En SPE/ICoTA Coiled Tubing & Well Intervention Conference & Exhibition. Society of Petroleum Engineers, 2015. DE LAAT, Joseph; GALLARD, HervÉ. Catalytic decomposition of hydrogen peroxide by Fe (III) in homogeneous aqueous solution: mechanism and kinetic modeling. Environmental Science & Technology, 1999, vol. 33, no 16, p. 2726-2732.


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HULL, Katherine L., et al. Recent Advances in Viscoelastic Surfactants for Improved Production from Hydrocarbon Reservoirs. En SPE International Symposium on Oilfield Chemistry. Society of Petroleum Engineers, 2015. KALFAYAN, Leonard John, et al. The Art and Practice of Acid Placement and Diversion: History, Present State, and Future. En SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, 2009. LIANG, Jenn-Tai, et al. Why do gels reduce water permeability more than oil permeability?. SPE Reservoir Engineering, 1995, vol. 10, no 04, p. 282-286. LIU, Ming, et al. Diverting mechanism of viscoelastic surfactant-based self-diverting acid and its simulation. Journal of Petroleum Science and Engineering, 2013, vol. 105, p. 91-99. SIDDIQUI, S., et al. An experimental investigation of the diversion characteristics of foam in Berea sandstone cores of contrasting permeabilities.Journal of Petroleum Science and Engineering, 2003, vol. 37, no 1, p. 51-67. SYDANSK, Robert D., et al. A newly developed chromium (III) gel technology.SPE Reservoir Engineering, 1990, vol. 5, no 03, p. 346-352. VENTRESCA, Maria Luisa; FERNANDEZ, Iraima; NAVARRO-PEREZ, Geidy.Reversible gelling system and method using same during well treatments. U.S. Patent No 7,994,100, 9 Ago. 2011. VERNÁEZ, Oscar, et al. Dynamic Rheology and Relaxation Time Spectra of Oil-Based Self-degradable Gels. Journal of Polymer Sciences: Polymer Physics, 2015a. to be published. VERNÁEZ, Oscar, et al. Degradation of styrene butadiene rubber (SBR) in anaerobic conditions. Polymer Degradation and Stability, 2015b, vol. 111, p. 159-168.. ZITHA, Pacelli LJ, et al. Control of flow through porous media using polymer gels. Journal of applied physics, 2002, vol. 92, no 2, p. 1143-1153.

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Overall Conclusion

The work presented in this thesis as a compilation of several papers corresponds to the development of an original product designed to be used in oil production operations. The new methodology for rheological characterization of an oil-based self-degradable gel had to be developed because the frequency ranges of the experimental data that could be obtained were constraint to a few decades. The development of this method was challenging because it implied the resolution of an ill-posed problem, and also because, regardless the availability of several methods found in literature, only one had studied the effect of short frequency range data. The KKT method proposed was successfully validated not only against real experimental data but also against a simulated spectrum normally used to validate the different methods available in literature. This method allows obtaining a continuous relaxation time spectrum which shows that the relaxation modes can be related to structural changes in the sample evaluated by in-situ rheological measurements. With this tool, the rheological characterization of the system was possible and the liquid-solid transitions during crosslinking and degradation were observed. Once a rheological characterization method was available, the effect of the different formulation parameters such as the concentration of polymer, crosslinker and degradation agents could be evaluated. The gelation time, strength and degradation time can be controlled by adjusting the concentration of the reactants. The polymer system experiments a liquid-solid transition and structural changes to form a polymer network with both physical entanglements and chemical crosslinks. Adding hydroperoxide to the initial formulation degrades the polymer chains as they simultaneously crosslink. The resulting rheological behavior is the consequence of a competition of two simultaneous reactions, crosslinking and degradation, and both can be retarded by adding an scavenger, which in this case an alkylbencene was used.

Overall Conclusion

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To evaluate the degradation reactions, knowing that the concentration of chemical changes was not enough to use conventional spectroscopic methods, the changes in molecular weight distributions were characterized. Then, the population balance equation analysis was an important tool to quantify the degradation process, and even with high experimental errors, it was concluded that the main degradation mechanism for the SBR degradation in solution under anaerobic conditions was random chain scission which are dependent on the molecular weight. The macro-radicals of polymer chains generate new smaller chains by chain scission processes. The alkylbencene retards the degradation reaction of SBR in solution, as it does for crosslinking and degradation reaction in the gel system. Then, one major finding in the development of this work was the use of this common solvent for controlling the gelation and degradation reactions. Finally, the results obtained in core flood test were found to be consistent with all the previous studies, as rheological behavior, where maximum gel strength defines the pressure resistance of the diverting agent. Hence, it is possible to design a tailor made formulation for different operational need. Since the diversion and productivity recovery tests were successful, applying the scaling rules mentioned in the last chapter makes possible to design a formulation ready to be use in a pilot field test. From the field test, other parameters will be evaluated to tune up the formulation.

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