Chapitre 2 : Rhéologie

Fluides complexes

N.Vandewalle, Professeur Ordinaire, Université de Liège

1. Elasticité-plasticité

2. Ecoulements

3. Rhéologie

4. Hysteresis

5. Rhéométrie

6. Viscoélasticité

7. Rhéométrie dynamique

Plan du chapitre

8. Modèles phénoménologiques

1. Elasticité-plasticité• Trois types de déformation :

x

y

� = F/S

��

d

u

� solide

y

solide

x

p

p

p

p

�V

y

solide

x

� = F/s��

�`

� = Gu

d

�p = �K�V

V0

� = E�`

`0

compressibilité

module de cisaillement

module de Young

• Cisaillement d’un solide : loi de Hooke

� =u

d

⇥ = G�

x

y

� = F/S

��

d

u

� solide

[déformation]

[loi de Hooke]

• Module de cisaillement : - polyéthylène- glace- verre- diamant

G = 108 PaG = 109 PaG = 1011 PaG = 1012 Pa

1

E=

1

9K+

1

3G

[You

ng]

[com

pres

sibi

lité]

[cis

aille

men

t]

• Rupture : contrainte seuil (ou déformation seuil) à dépasser�

�

�c

• Propagation de cracks :

fractures régulières et irrégulières

• Plasticité : cisaillement continu

⇥ � ⇥c ⇥ (�)1/n

⇥ � ⇥c ⇥ ln �

�

�c

réversible irréversible � = t�

• Origine de la plasticité : déplacement des défauts, tapis de Mott

Il est plus facile de déplacer un pli dans un tapis que le tapis lui-même.

� = cte

• Transition fragile-ductile : activation du mouvement des dislocations

USS Ponaganset s’est fracturé en deux parties dans les eaux froides de Boston en 1947.

- ductile (mou) à haute température

- fragile (cassant) à basse température

- exemple : naufrage du Titanic et bateaux de la WW2

(T > Tc)

(T < Tc)

x

y

� = F/S

��

dfluide

⇥v

⇤ = ⇥�

[taux de cisaillement]

[viscosité]

� =v

d

• Viscosités : - air

- eau

- huile silicone

� = 10�3 Pa s� = 210�5 Pa s

� = 10�3 � 10+3 Pa s

2. Ecoulements• Fluide Newtonien :

• La plus longue expérience du monde : gouttes de bitume

«pitch drop experiment»

- démarrée en 1927 - toujours en cours- à l’Université du Queensland (Brisbane)- liquide = bitume- 8 gouttes sont tombées depuis 1927- bientôt la 9ème !

1 1938

2 1947

3 1954

4 1962

5 1970

6 1979

7 1988

8 2000

• Cisaillement continu : � = cte

�

� = t�

visqueux

plastique

élasti

que

• Effet de la température : élasticité et viscosité diminuent avec T

�

� = t�

T �

T � � ⇥ exp��T0

T

⇥

Arrhenius

comparaison entre comportement solide et liquide

• Effet de la température : le cas de l’eau

La viscosité de l’eau varie presque d’un facteur 10 entre 0°C et 100°C !

3. Rhéologie• La plupart des fluides complexes sont non-newtoniens

• Quatre grands types d’écoulement :

�

newtonien

rhéofluidifiant

rhéoépaississant fluide à seuil

�

cfr détails sur les diapositives suivantes...

� = k�n (n < 1)

� = k�n (n > 1)� � �s = ⌘�

[Bingham]

[Ostwald]

fluide rhéofluidifiant[non Newtonien]

• Fluides rhéofluidifiants et rhéoépaississants :

fluide visqueux [Newtonien]

fluide rhéoépaississant[non Newtonien]

�

�

�v

• Exemple : suspensions dures dans un liquide

différentes concentrations de billes

• Autres exemples : polymères et sang

⌘ � ⌘1⌘0 � ⌘1

= [1 + (��)a]n�1a

• Polymère rhéoépaississant

http://www.youtube.com/watch?v=9k9baW_UOsw

• Fluides à seuil :

�

�

�s

solide

fluide

- Exemples : peinture, pâte dentifrice, chocolat fondu, suspensions floculées, miracle du sang de Saint Janvier (Naples), ...

- Modèles : ⇤ � ⇤s

⇥= � (Bingham)

⇥ � ⇥s = k�n

⇥⇤ �

⇥⇤s =

�⇥�

(Herschel-Bulkley)

(Casson)

4. Hysteresis• Thixotropie : la viscosité dépend de l’histoire ‘AB’ du fluide

�

�

thixotropie

A

B

�

�

antithixotropie

B

A

• Cela complique la rhéométrie du fluide complexe (durée des mesures) !

- béton- mousse à raser- boues- ketchup

(rhéopexie)

Non-Newtonian Fluids: An Introduction 15

Fig. 9 Typical experimental data showing thixotropic behaviour in a red mud suspension [33]

4.1 Thixotropic Behaviour

A material is classified as being thixotropic if, when it is sheared at a constant rate,its apparent viscosity η = σ/γ (or the value of σ because γ is constant) decreaseswith the duration of shearing, as shown in Fig. 9 for a red mud suspension con-taining 59% (by wt) solids [33]. As the value of γ is gradually increased, the timeneeded to reach the equilibrium value of σ is seen to drop dramatically. For instance,at γ = 3.5 s!1, it is of the order of " 1500 s which drops to the value of " 500 sat γ = 56 s!1. Conversely, if the flow curve of such a fluid is measured in a singleexperiment in which the value of γ is steadily increased at a constant rate from zeroto some maximum value and then decreased at the same rate, a hysteresis loop ofthe form shown schematically in Fig. 10 is obtained. Naturally, the height, shapeand the area enclosed by the loop depend on the experimental conditions like therate of increase/decrease of shear rate, the maximum value of shear rate, and thepast kinematic history of the sample. It stands to reason that, the larger the enclosedarea, more severe is the time-dependent behavior of the material under discussion.Evidently, the enclosed area would be zero for a purely viscous fluid, i.e., no hys-teresis effect is expected for time-independent fluids. Data for a cement paste [38]shown in Fig. 11 confirms its thixotropic behavior. Furthermore, in some cases, thebreakdown of structure may be reversible, i.e., upon removal of the external shearand following a long period of rest, the fluid may regain (rebuilding of structure)the initial value of viscosity. The data for a lotion shown in Fig. 12 illustrates thisaspect of thixotropy. Here, the apparent viscosity is seen to drop from " 80 Pa.s

16 R.P. Chhabra

to ! 10 Pa.s in about 5" 10 s when sheared at γ = 100 s"1 and upon removal ofthe shear, it builds up to almost its initial value in about 50" 60 s. Barnes [2] haswritten a thorough review of thixotropic behavior encountered in scores of systemsof industrial significance.

Fig. 10 Qualitative shearstress-shear rate behaviourfor thixotropic and rheopecticmaterials

Fig. 11 Thixotropy in acement paste

• Exemples :

Thixotropie du ciment : montée et descente du taux de cisaillement

Evolution temporelle de la contrainte de cisaillement pour des boues,

à différents taux de cisaillement.

géométrie cône-plan géométrie de Couette

�,⇥

�0

R

⇥ =3�

2�R3

� =�⇥0

� =3⇥0

2⇤R3

�⇥

�,⇥

�0

R1 R2

h

⇥ =�

2�hR21

� =R1�

R2 �R1

� =R2 �R1

2⇥hR31

�⇥

5. Rhéométrie

8 R.P. Chhabra

In most cases, the value of η∞ is only slightly higher than the solvent viscosity ηs.Fig. 5 shows this behavior in a polymer solution embracing the full spectrum ofvalues going from η0 to η∞ . Obviously, the infinite-shear limit is not seen for poly-mer melts and blends, or foams or emulsions or suspensions. Thus, the apparentviscosity of a pseudoplastic substance decreases with the increasing shear rate, asshown in Fig. 6 for three polymer solutions where not only the values of η0 are seento be different in each case, but the rate of decrease of viscosity with shear rate isalso seen to vary from one system to another as well as with the shear rate intervalconsidered. Lastly, the value of shear rate marking the onset of shear-thinning isinfluenced by several factors such as the nature and concentration of polymer, thenature of solvent, etc for polymer solutions and particle size shape, concentrationof solids in suspensions, for instance. Therefore, it is impossible to suggest validgeneralizations, but many polymeric systems exhibit the zero-shear viscosity regionbelow γ < 10!2 s!1. Usually, the zero-shear viscosity region expands as the molec-ular weight of polymer falls, or its molecular weight distribution becomes narrower,or as the concentration of polymer in the solution is reduced.

Fig. 5 Demonstration of zero shear and infinite shear viscosities for a polymer solution

The next question which immediately comes to mind is that how do we approxi-mate this type of fluid behavior? Over the past 100 years or so, many mathematicalequations of varying complexity and forms have been reported in the literature;some of these are straightforward attempts at curve fitting the experimental data(σ ! γ) while others have some theoretical basis (blended with empiricism) in sta-tistical mechanics as an extension of the application of kinetic theory to the liquidstate, etc. [9]. While extensive listing of viscosity models is available in several

• Attention, chaque rhéomètre possède sa gamme de sensibilité :

• Fluides viscoélastiques :

- Maxwell remarque que les fluides sont à la fois visqueux et élastiques.

- Tout est une question d’échelle de temps.

⇥ = G�t

⇤ = ⇥�

⇥ =�

G

�

t

élasti

que

visqueux

�mémoire de la forme initiale

- Temps caractéristique de l’eau : ⇥ =�

G� 10�3 Pa s

109 Pa= 10�12 s

module de la glace

6. Viscoélasticité

http://www.youtube.com/watch?v=nX6GxoiCneY

• Trois effets viscoélastiques : Barus, Weissenberg, Kaye

http://www.maniacworld.com/The-Kaye-Effect.html

Fig. 6 – Siphon ouvert realise avec une solution aqueuse de 0,75% de Polyox WSR 301 (photogra-phies tirees de [2]). La sequence montre le developpement du siphon a partir du versement initialhors du becher.

Fig. 7 – Detente elastique (photographies tirees de [2]).

7

• Siphon d’un fluide élastique :

• Détente élastique :

Fig. 6 – Siphon ouvert realise avec une solution aqueuse de 0,75% de Polyox WSR 301 (photogra-phies tirees de [2]). La sequence montre le developpement du siphon a partir du versement initialhors du becher.

Fig. 7 – Detente elastique (photographies tirees de [2]).

7

• Classer les fluides selon leur temps de relaxation viscoélastique

�0 s 10�12 s 10�3 s 103 s

liquide parfait

eau polymères émulsions?

liquides visqueux linéaires

(Newtonien)(� = 0)

liquides visqueux non-linéaires

(non-Newtonien)

matièresviscoélastiques

Microgel Physics

Microgels are small particles made of crosslinkedpolymer usually suspended in a liquid. Their interesting properties derive by the possibility of drastically changing the quality of the interactions between the polymer and the suspending medium in response to small changes of temperature or solution PH. To see how these properties may be applied such as in controlled release, in situ reaction…please look at the JinwoongKim and Rhutesh Shah webpages. I’m trying to use them as a model system for colloidal suspensions, to study general problems ranging from the glass-like to the particle-gel phase and dynamics. To this aim with the help of Jinwoong, we synthesized fluorescent visible particles at different cross-linker concentrations (see side figure). In this way we can take advantage of confocal microscopy in addition to rheology and light scattering to have more insights in these fascinating fields of physics.

Top: confocal image of

microgel particles on the glass.

Bottom: solution at 0.01% of

microgel particles in water and

acrilic acid .

Web page maintained by Giovanni Romeo

Collaborators: Alberto Fernandez-Nieves, Hans Wyss, Jinwoong Kim

7. Rhéométrie dynamique• Principe : oscillations sinusoïdales de la déformation

t

�

�

t

élastique

visqueuxviscoélastique

�

� = �0 sin(!t)

� = �0! cos(!t)

- pas de déphasage : solide élastique :

- déphasage de pi/2 : liquide newtonien :

- déphasage quelconque : système viscoélastique :

� = �0 sin(!t+ �)

� = G� = G�0 sin(!t)

� = ⌘� = ⌘�0! sin(!t+ ⇡/2)

� = �0 sin(!t+ �)

• Module de cisaillement complexe :

� = �0 exp(i!t) � = �0 exp (i(!t+ �))

� = [G0 + iG00] � tan � =G00

G0

composante élastique

composante visqueuse

104 ANNEXE A. FLUIDES NON NEWTONIENS

Fig. A.10 – Modules de cisaillement pour une serie de polystyrenes lineaires de masses moleculairesallant de 8900 (L9) a 580000 (L18). D’apres S. Onogi, T. Masuda, K. Kitagawa, Macromolecules3, 109 (1970)

Fig. A.11 – Contraintes engendrees par un ecoulement de cisaillement.

104 ANNEXE A. FLUIDES NON NEWTONIENS

Fig. A.10 – Modules de cisaillement pour une serie de polystyrenes lineaires de masses moleculairesallant de 8900 (L9) a 580000 (L18). D’apres S. Onogi, T. Masuda, K. Kitagawa, Macromolecules3, 109 (1970)

Fig. A.11 – Contraintes engendrees par un ecoulement de cisaillement.

solutions de polymères fondus

• Modèle phénoménologique de Maxwell :

�1 = G�1

�2 = ⌘�2 � = �1 = �2

� = �1 + �2

� = �1 + �2 � =�1

G+

�2

⌘

� + ⌧ � = ⌘� ⌧ =⌘

Gavec

- composantes élastiques et visqueuses- composantes en série

8. Modèles phénoménologiques

• Test de relaxation :� = �0 (t > t0)� = 0 (t < t0) � = G0�0 (t = t0)

t

�

t0

�0

t

G G0

� + ⌧ � = ⌘�

� + ⌧ � = 0

G(t) =�

�0= G0 exp

✓� t

⌧

◆

la ‘mémoire’ disparaît sur un temps caractéristique tau

⌧

• Cisaillement dynamique dans le modèle de Maxwell :

G0 = G!2⌧2

1 + !2⌧2

G00 = G!⌧

1 + !2⌧2

� + ⌧ � = ⌘�

recherche de solutions oscillantes

composante élastique sature aux hautes fréquences

composante visqueuse présente un maximum en ! =1

⌧

ln!

lnG0

lnG00ln 1/⌧

40

Do Maxwell Fluids exist?

Water oil emulsionConcentrated surfactant solution

Do Maxwell fluids exist? Well at least some fluids do behave according to the simple

maxwell model, at least to a good approximation. These are fluids in which one very

dominant relaxation mechanism exists. Two examples are given above:

• Emulsions : here the VE comes from the interface, the material now can store

elastic energy as the droplets, which have an interfacial tension, can deform. The

relaxation time of these fluids is determined by the medium viscosity, the droplet size

and the value of the interfacial tension. When the droplets are fairly uniform in size,

the system will respond with essentially one single relation time. The graph on the left

compares two emulsions which were subjected to different mixing histories.

• Surfactants : A lot of shampoos come relatively close to being Maxwell fluids! Such

surfactants can form - in a certain range of concentration - rodlike micellar objects.

These objects are sometimes called “Living Polymers” as they resemble polymeric

fluids but they can break and reform without too much effort. The characteristic time

scale of these systems is the breaking or reconnection time.

40

Do Maxwell Fluids exist?

Water oil emulsion Concentrated surfactant solution

Do Maxwell fluids exist? Well at least some fluids do behave according to the simplemaxwell model, at least to a good approximation. These are fluids in which one verydominant relaxation mechanism exists. Two examples are given above:

• Emulsions : here the VE comes from the interface, the material now can storeelastic energy as the droplets, which have an interfacial tension, can deform. Therelaxation time of these fluids is determined by the medium viscosity, the droplet sizeand the value of the interfacial tension. When the droplets are fairly uniform in size,the system will respond with essentially one single relation time. The graph on the leftcompares two emulsions which were subjected to different mixing histories.

• Surfactants : A lot of shampoos come relatively close to being Maxwell fluids! Suchsurfactants can form - in a certain range of concentration - rodlike micellar objects.These objects are sometimes called “Living Polymers” as they resemble polymericfluids but they can break and reform without too much effort. The characteristic timescale of these systems is the breaking or reconnection time.

solution concentrée de surfactantsémulsion eau-huile

• Modèle phénoménologique de Kelvin-Voigt :

�1 = G�1

�2 = ⌘�2

- composantes élastiques et visqueuses- composantes en parallèle

� = �1 = �2

� = �1 + �2

� = G� + ⌘�

• Test de fluage :

tt0

t

�

�

�0

� =

�0

G

1� exp

✓� t

⌧

◆�

� = G� + ⌘�

⌧

• Cisaillement dynamique dans le modèle de Kelvin-Voigt : � = G� + ⌘�

recherche de solutions oscillantes

composante élastique constante, domine aux temps long (élasticité retardée)

composante visqueuse

G0 = G

G00 = ⌘!

ln!

lnG0

lnG00