RHEOLOGY OF COMPLEX FLUIDS PART 1 WORMLIKE MICELLES O. Manero Instituto de Investigaciones en Materiales Facultad de Química UNAM

WHAT IS SHEAR-BANDING FLOW ? It is a multi-valued region between and caused by mechanical instabilities and/or a shear-induced first- order phase transition Two fluid bands coexist supporting two different shear rates

Cappelaere et al. PRE, 56, 1997, POLYMER-LIKE MICELLES OF CTAB-D 2 O Structural transition under flow: the simple view

SHEAR BANDING FLOW HAS BEEN OBSERVED IN: + MONODISPERSED POLYMER MELTS AND SOLUTIONS + POLYMER-LIKE MICELLES + COLLODAL CRYSTALS + THERMOTROPIC LIQUID CRYSTALLINE POLYMERS + LYOTROPIC LIQUID CRYSTALS + DIBLOCK COPOLYMER MICOEMULSIONS OFTEN, A SHEAR-INDUCED PHASE TRANSITION OCCURS

SHEAR BANDING FLOW (SPURT EFFECT) WAS FIRST OBSERVED IN MONODISPERSED POLYMER MELTS Vinogradov, G. V. Rheol. Acta, 1973, 12, 357

Shear Banding Flow in Polymeric Bicontinuous Microemulsion Lodge and Bates’ Group, UMN

Eiser et al. PRE, 61, 6759, 2000

POLYMER-LIKE MICELLES  LONG FLEXIBLE RODS FORMED BY SURFACTANT MOLECULES  THEY CAN FORM ENTANGLEMENTS SIMILAR TO THOSE FORM IN POLYMER SOLUTIONS VISCOELASTIC SOLUTIONS

SHEAR BANDING FLOW IN POLYMER-LIKE (WORM-LIKE) MICELLAR SOLUTIONS MICELLAR GROWING: FROM SPHERES TO RODS POLYMER-LIKE MICELAR SOLUTIONS AS MODEL SYSTEMS FOR SHEAR-BANDING RELAXATION BEHAVIOR AND SHEAR BANDING FLOW MECHANICAL INSTABILITIES VERSUS SHEAR-INDUCED FIRST ORDER PHASE TRANSITION MODELLING NON-LINEAR RHEOLOGY

Å Å cmc INCREASING CONCENTRATION

SHEAR BANDING FLOW IN POLYMER-LIKE (WORM-LIKE) MICELLAR SOLUTIONS MICELLAR GROWING: FROM SPHERES TO RODS POLYMER-LIKE MICELAR SOLUTIONS AS MODEL SYSTEMS FOR SHEAR-BANDING RELAXATION BEHAVIOR AND SHEAR BANDING FLOW MECHANICAL INSTABILITIES VERSUS SHEAR-INDUCED FIRST ORDER PHASE TRANSITION MODELLING NON-LINEAR RHEOLOGY

Linear Viscoelastic Behavior bb  rep <<1 bb  rep 11 Near Maxwell behavior Polymer-like behavior Polymer-like micelar solutions usually have a wide size polydispersity

Non-linear viscoelastic behavior Long transients and oscillations At least two main relaxation time Shear banding pipe flow

Shear-Banding in Steady Shear Flow: An Schematic Representation Two homogeneous Newtonian regions (I and VI) A shear-thinning region (II) Two metastable regions (III and V). An unstable region (IV) leading to shear banding flow:  ( s ) Modulus G ' G "  d  R   c2   c1   Log 

Shear-Banding in Steady Shear Flow: Correlation with experimental data Two homogeneous Newtonian regions (I and VI) A shear-thinning region (II) Two metastable regions (III and V). An unstable region (IV) leading to shear banding flow. System: CTAT/H 2 O

SHEAR BANDING FLOW IN POLYMER-LIKE (WORM-LIKE) MICELLAR SOLUTIONS WHY THEY ARE MODEL SYSTEMS FOR SHEAR- BANDING? RELAXATION BEHAVIOR AND SHEAR BANDING FLOW MECHANICAL INSTABILITIES VERSUS SHEAR-INDUCED FIRST ORDER PHASE TRANSITION

Linear viscoelastic behavior IT IS POSSIBLE TO SHIFT FROM KINETIC-CONTROLLED TO DIFUSSION-CONTROLLED RELAXATION WHAT IS THE EFFECT ON NON-LINEAR RHEOLGY?

Effect of Temperature FAST BREAKING SLOW BREAKING Shear banding tends to vanish!! System: CTAT/H 2 O

Effect of Concentration KINETIC-CONTROLLED DIFUSSION-CONTROLLED Shear banding tends to fade!! System: CTAT/H 2 O

Effect of Ratio Salt/Surfactant FAST BREAKING SLOW BREAKING Shear banding tends to vanish!! System: DTAB/NaSal/H 2 O

SHEAR BANDING FLOWHOMOGENEOUS FLOW DISCONTINUOUS FLOW ( ) MONOTONIC FLOW ( ) FAST BREAKINGSLOW BREAKING SINGLE RELAXATION TIME SPECTRA OF RELAXATION TIMES

MODELING LinearNon-Linear Behavior Granek and Cates, J Chem Phys 96:4758 (1992) Bautista et. al., J. Non-New. Fluid Mech. 94, 57 (2000)

SHEAR BANDING FLOW IN POLYMER-LIKE (WORM-LIKE) MICELLAR SOLUTIONS WHY THEY ARE MODEL SYSTEMS FOR SHEAR- BANDING? RELAXATION BEHAVIOR AND SHEAR BANDING FLOW MECHANICAL INSTABILITIES VERSUS SHEAR-INDUCED FIRST ORDER PHASE TRANSITION

IS IT POSSIBLE TO OBTAIN A MASTER CURVE FROM NON-LINEAR RHEOLOGICAL DATA? YES! Porte et al., J. Phys II France (1997)

Master curve diagram for living polymers  (6 % - 12 %) semidilute/concentrated Berret et al. PRE, 55, 1997, 1668.

A B C D E

Stress Relaxation At the low-shear Newtonian region, the stress relaxation is single-exponential. But at (shear- banding region), the stress exhibits two relaxation mechanisms. Note that the stress relaxation curves approach a saturation value. At the high-shear rate Newtonian region, the relaxation is again single- exponential. C DTAB = 12 mM with C salt /C DTAB = 0.84 A B C A C B

Shear banding close to an I-N transition CTAB/D 2 O: 2 H spectroscopy across a Couette cell gap Fisher and Callagham, Europhysics Letters, 50, 803 (2000) Physical Review E, 6401, 1501 (2001) Shear rate = 700 s -1 Inner cilynder Fluid Housing Spacer Driving abat Isotropic component Ordered component Inner Wall Outer Wall Distance from centre (mm)

SHEAR-BANDING FLOW AS MECHANICAL INSTABILITY SCARC, IF ANY, EXPERIMENTAL EVIDENCE BASICALLY, THE MECHANICAL INSTABILITY HAS BEEN USED AS SUPPORTING ARGUMENT OF SHEAR-BANDING MODELLING Spenley et al., J. Phys. II France (1996). Greco et al., J. Non-Newtonian Fluid Mechs.(1997) Larson, R. G. Rheol. Acta 1992 Byars et al., J. Fluid Mech. 1994

SHEAR BANDING FLOW IN POLYMER-LIKE (WORM-LIKE) MICELLAR SOLUTIONS MICELLAR GROWING: FROM SPHERES TO RODS POLYMER-LIKE MICELAR SOLUTIONS AS MODEL SYSTEMS FOR SHEAR-BANDING RELAXATION BEHAVIOR AND SHEAR BANDING FLOW MECHANICAL INSTABILITIES VERSUS SHEAR-INDUCED FIRST ORDER PHASE TRANSITION MODELLING NON-LINEAR RHEOLOGY

 o First Newtonian fluidity   Second Newtonian fluidity Characteristic structure time  Shear banding intensity parameter k 0 Kinetic constant for scission  1 Shear banding intensity parameter  All parameters depend on temperature and concentration.  The steady state solution is MODELLING NON-LINEAR RHEOLOGY Bautista et al., J. Phys. Chem (2002).

Shear-banding in steady shear flow schematical representation

Predictions of the model I. Homogeneous flow region II. Metastable flow regions III. Heterogeneous (“spinodal”) flow region  As   increases, the shear banding region becomes wider EFFECT OF  1

SHEAR INDUCED PHASE TRANSITIONS  The coexistence region diminish with temperature  The Newtonian region shifts to high shear rates with temperature.  There is an unstability region limited by a spinodal curve

Non-linear viscoelasticity Shear Banding Flow Experimental data (Escalante et al., Langmuir (2003)) are predicted successfully with our model.

Stress relaxation with two characteristic times Long transients and oscilations Bautista et al., J. Non-Newtonian Fluid Mechs. (2000)

Steady State vs. Transient Profiles Bautista et al., J. Phys. Chem. (2002)

Shear banding and Metastable states  Multi-valued region between and  How is the stress plateau chosen?

EXTENDED IRREVERSIBLE THERMODYNAMIC ANALYSIS where Bautista et al. J Phys Chem (2002)

IRREVERSIBLE THERMODYNAMICS FLOW ANALISYS: RESULTS  There are two homogeneous regions  There is a multivalued region in shear rate.  The stability condition is violated in a given shear rate region.  Band coexistence is observed only when they have the same extended Gibbs free energy.  The equal area condition is valid only when normal stress can be neglected  The law of lever can be applied

COMPARISON WITH EXPERIMENTAL DATA  In the shear controlled data, there is a shear rate for each stress value, while in the stress controlled data the shear rate is multi valued.  Structure coexistence is presented when two points have the same free energy of Gibbs.  In the shear controlled mode the metastable states are exhibited.  The predictions agree with experimental data System: CTAT/H 2 O

PREDICTIONS OF THE MODEL Bautista et al. JNNFM 94, 57, 2000

MODEL RENORMALIZATION THE MASTER PHASE DIAGRAMA  FOR STEADY STATE CONDITIONS, OUR MODEL REDUCES TO: BUT SINCE  0 = (  0 ) -1 = (G 0  R ) -1, THEN EQ. (A) REDUCES TO: &: RENORMALIZED SHEAR STRESS AND RATE (A) (B) IT IS EASY TO SHOWN THAT EQ. (B) YIELDS:

All experimental data collapse into a master-curve at low shear rates with the renormalization:   (  /G 0 ) and     R. RENORMALIZATION OF STEADY SHEAR DATA THIS RENORMALIZATION IS PREDICTED BY OUR MODEL

CONCLUSIONS SHEAR BANDING FLOW OCCURS ONLY WHEN THE RELAXATION OF THE MICELLES IS KINETICALLY-CONTROLLED (FAST- BREAKING) SHEAR-BANDING FLOW IN POLYMER-LIKE MICELLAR SOLUTIONS APPEARS TO BE A FIRST-ORDER PHASE TRANSITION A SIMPLE MODEL CAN REPRODUCE MOST OF THE FEATURES OF THE NON-LINEAR RHEOLOGY OF THESE SYSTEMS

Small-angle light-scattering Butterfly patterns: concentration fluctuations that couple to the mechanical stress. They appear in the region where the stress is still increasing with shear rate. Nucleation takes place before the mechanical instability appears.

Modelling the flow-concentration coupling Helfand and Fredrickson model (Phys. Rev. Lett. 62 (1989) 2468). Structure Factor: Rouse dynamics where << Dixon et al experiment: a peak in S(k) due to the crossover of time scales: Phys.Rev.Lett. 66 (1991)2408, 68 (1992) 2239.

The transient gel Brochard & de Gennes, Macromolecules 10 (1997) It behaves as a simple two-fluid mixture on time scales long comparable to stress relaxation time, but at short time scales it behaves as a solvated gel. Two modes of relaxation of S(k): Experimentally proved in micellar systems (Kadoma et al, Phys.Rev.Lett. 76 (1996) 4432).

Extensions to the H-F Theory Milner (Phys.Rev.Lett. 66 (1991) 1477, Phys.Rev.E 48(1993)3674). Doi & Onuki (J.Phys. II 2 (1992) 1631).

A kinetic-thermodynamic model Extended Irreversible Thermodynamics. Applying the EIT procedure to the non- conserved variables:

At low shear rates These are of the same form as those of Milner and Doi, Onuki. HF is recovered when the relaxation time→zero. The two modes of a transient gel are predicted in the relaxation of S(k).

Conclusions Shear Banding is a manifestation of the interactions of two mechanisms: -Mechanical or constitutive instabilities -Shear induced structures (first-order phase transition) Modelling: Kinetic approach + reptation of wormlike micelles Stress-concentration coupling A kinetic-thermodynamic model is proposed