Rheology of Dispersions

© BASF AG

Outline • Hard particles • Interactions among colloidal particles • Repulsive particles • Particle size distribution • Shear thickening • Attractive particles

Prof. Dr. N. Willenbacher Institute for Mechanical Process Engineering and Mechanics

Rheology H. Barnes, B. Hutton, K. Walters, Introduction to Rheology C. Macosko, Rheology: Principles, Measurements and Applications R. Larson, The Structure and Rheology of Complex Fluids N.J. Wagner and J. Mewis Colloidal Dispersion Rheology

Colloids D. Fennel Evans, H. Wennerström The Colloidal Domain W. B. Russel, D. A. Saville, W. R. Schowalter Colloidal Dispersions T. Tadros (Ed.) Colloid Stability Volume 1: The Role of Surface Forces Part I: Colloids and Interface Science J. Goodwin Colloids and Interfaces with Surfactants and Polymers - An Introduction

Textbooks

Parameters Controlling Dispersion Rheology

d i r e c t e d f l o w

v →

• Particle volume fraction φ

• Particle size sphere radius a shape size distribution

• Brownian motion • Particle interactions hydrodynamic / shear forces thermodynamic electrostatic repulsion

steric van der Waals attraction depletion

s Brownian Motion

2

3

2

3

6

6

πη

τ

τ γ

ηπ γ

γ

σσ

=

=

= ⋅

=

=

=

B

B

r

kTDa

aD

PeakT

aD

akT

Stokes – Einstein no Brownian motion if viscosity is high and/or particles are large

characteristic time scale

Peclet number dimensionless shear rate

dimensionless stress

collisions with solvent molecules stochastic force, random motion prevents particles from settling

fat-droplets in skim-milk

Pe → ∞ shear forces dominate Pe → 0 Brownian motion dominates

s Hard Spheres

a

2 a d i s t a n c e

Ψ ∞

• Brownian motion

• Hydrodynamic interactions

the term "hard" refers to the shape of the interaction potential, not only solid particles, but also liquid droplets or even gas bubbles can be treated as "hard" spheres

Maximum Packing Fraction

monodisperse ellipsoids and other irregular shaped objects pack closer

φmax = 0.74 for random packing

random body centered cubic face centered cubic

φmax = 0.63 = 0.68 = 0.74

Weitz, Science 2004

Hard Sphere Phase Diagram

v o l u m e f r a c t i o n φ 0 0 . 5 0 . 5 5 0 . 5 8 0 . 6 3

s o l i d l i q u i d c r y s t a l l i n e l i q u i d

c o e x i s t e n c e

c r y s t a l - l i n e*

g l a s s y r a n d o m c l o s e

p a c k i n g

0.74

fcc crystal

PMMA particles in organic solvent

Pusey & van Megen Nature

*) crystallization takes place since gain of volume entropy dominates loss of configurational entropy !

„freezing“ of diffusion processes / particle mobility analogous to glass transition in small molecule and polymeric glasses

control parameter φ instead of T !

φ << φg φ ≈ φg

cage effect

Glass Transition in Dispersions

for colloidal suspensions

viscosity diverges at φg 0 1

− = − S g

Aγ

η φη φ

γ = 1.6 Krieger-Dougherty = 2 Quemada = 2.55 Mode coupling theory

1 0 4

1 0 3

1 0 2

1 0 1

1 0 0

φ / φ m a x

1 . 0 0 . 8 0 . 6 0 . 4 0 . 2 0

η r = η 0 / η s

E i n s t e i n

B a t c h e l o r

K r i e g e r - D o u g h e r t y

Q u e m a d a

Q u e m a d a

1 – φ

φ m a x

- 2

η r =

K r i e g e r - D o u g h e r t y

1 – φ

φ m a x

- 2 . 5 φ m a x

η r = r e d u c e s t o E i n s t e i n a s φ → 0

B a t c h e l o r η r = 1 + 2 . 5 φ + 6 . 2 φ 2 c o r r e c t f o r

φ < 0 . 1

E i n s t e i n η r = 1 + 2 . 5 φ c o r r e c t f o r

φ < 0 . 0 1

Zero Shear Viscosity vs. Volume Fraction

Hard Spheres – Viscosity vs. Shear Stress

particle radius

fluid viscosity

I.M Krieger, in Polymer Colloids, 1988

PMMA particles in silicon oil

rη

0.45

0.40

0.30

φ

φ

φ

=

=

=0.20φ =0.10φ =

0

11 ( )

η ηη η φ σ

∞

∞

−=

− + rb

00

σ η ησ η η∞

= → =→ ∞ → =

r

r

Low shear viscosity increases stronger with φ than high shear viscosity

3 0

2 0

1 5

1 0

0 1 0 - 2

η r

1 0 - 1 1 0 0 1 0 1 1 0 2

2 5

η 0 , r

η ∞ , r

γ / s - 1

3 0

2 0

1 5

5

0 1 0 - 3

η r

1 0 - 1 1 0 0 1 0 1 1 0 3

2 5

1 0 - 2 1 0 2

C h o i + K r i e g e r , J C I S 1 9 8 6

a 3 η

k T P e = γ

φ = 0 . 4 5

8 5 n m 1 4 1 n m 3 1 0 n m

1 0

d e c r e a s i n g p a r t i c l e s i z e

Hard Spheres – Viscosity & Particle Size

The viscosity of hard sphres is independent of particle size, the onset of shear thinning is shifted to higher shear rates as the particle size decreases

101

100

10-2

η [Pa⋅s]

10-1 100 101 102

25

20

15

1010-3

ηr = η0 / ηs

10-1 100 10110-2

γ [s-1]

102

Krieger, Adv. Coll. Interface Sci. 1972

φ = 0.5

waterbenzylalcoholm-cresol

PS latex in

Pe =a3 η∞

kTγ

increasingsolvent viscosity

Hard-sphere Suspensions – Effect of Solvent ViscosityHard Spheres – Effect of Solvent Viscosity

·

φ = 0.5

viscosity

shear rate

10-2 10-1 100 101 102100

101

102

103

shear rate / a.u.

viscosity / a.u.

increasing solvent viscosity

s

s Hard Rods and Disks - 1 Axisymmetric particles

Parb

=2a

prolate / rodlike (a>>b): glass, graphite fibers, viruses, proteins, DNA

oblate / disklike (a<<b): red blood cells, mica flakes, clay

Orientation distribution

controlled by balance between hydrodynamic and Brownian forces

axis ratio 2b

random orientation at low shear rates for small particles and low fluid viscosity

flow alignment at high shear rates for large particles and high fluid viscosity

Hard Rods and Disks – 2 22

*

3

243

b a baa

πφ π ≈ ≈

rod

*φ critical volume fraction at which particles start to interact much smaller for non-spherical particles than for spheres

maxφ larger for non-spherical particles than for spheres

dilute semi-dilute

concentrated isotropic

nematic

*φ

Non-Spherical Particles at High Volume Fraction

Giesekus 1983 Clarke 1967

10 20 30 40 0 20 30 40 particle volume fraction / %

v

glass rods 30 x 700 mm

glass plates 100 x 400 mm

quartz grains 50 x 70 mm

spheres 40 mm glass fibres

21LD =

14LD =

7LD =

1LD =

0 10 20 30 particle volume fraction / %

Low shear viscosity increases with increasing anisotropy (at const. φ)

For anisotropic particles random orientation leads to a higher barrier to start flow, i.e. to an increase in low shear viscosity. However, under shear, these elongated particles can orient in the direction of flow, resulting in a lower high shear viscosity than for spherical particles with equivalent size.

Anisotropic Particle Suspensions at High Shear Rates

0,1 0,2 0,3 0,4 0,5 0,60,01

0,1

1

φ

h/2a

s

max3/ 2 1h a φφ

= −

φ = 0.5

2a = 100 nm h = 8 nm

2a = 10 µm h = 800 nm

at constant volume fraction distance between particle surface decreases with particle size

Colloidal Interactions

r

2a h

• orientation averaged dipole-dipole interaction Keesom • dipole-induced dipole interaction Debye • fluctuating dipole-induced dipole interaction London

van der Waals forces originate from electrostatic dipole-dipole interactions

van der Waals Attraction

2 2 2 2

2 2 2 2

1 2 2 4 ln6 4vdW H

a a r aAr a r r

−Ψ = − + + −

two spherical particles with radius a and separation r

12H

vdWA a

hΨ ≈ −

Derjaguin approximation valid for h<<a (small gap between particles)

summation and thermal averaging over all molecular dipoles

AH = Hamaker constant controlled by dielectric properties of particles + surrounding fluid dielectric constant ε and refractive index nA,B

typical value AH » 10-20 J typical range of vdW interaction 5 – 10 nm

+ +

+ +

+ + +

+

+

+

+

+

+

+

+

+

+

char

ge d

ensi

ty ρ

colloidal particle - - - - - - - - - - - -

particle surface solvent

Surface Charge & Electrostatic Double Layer

electrostatic potential around a charged particle in a dispersion decays exponentially due to shielding effect of counter-ions

1( ) exp( )el r rr

κΨ −

r i iiwith kT e n z

1/ 22 20 / = ∑κ ε ε

Debye length κ-1 "range of electrostatic repulsion"

s Steric Interaction

L

a

L = thickness of stabilizing layer φp = polymer concentration in stabilizing layer

χ = Flory-Huggins parameter

ν1 = volume of a solvent molecule

h = r-2a gap between particles

approximation for thin stabilizing layer (L<<a)

22

1

2 2

1

0 2

4 1 22 2

4 1 1 ln2 2 4

π φ χν

π φ χν

Ψ= ≤

Ψ = − − ≤ <

Ψ = − − − <

steric

stericp

stericp

L hkT

a hL L h LkT

a h hL h LkT L L

1 021 021 02

χ

χ

χ

< Ψ >

= Ψ =

> Ψ <

good solvent

poor solvent

Q - solvent

repulsive

attractive Napper, J Colloid Interface Sci, 1977

adsorbed, grafted or co-polymerized polymer chains on particle surface excess polymer concentration in the overlap region creates osmotic pressure → repulsive interaction

s DLVO Potential

2a

Ψ (r

)

r

r

2a

ΨvdW (r)

Ψel (r) energy barrier

( ) ( ) ( )vdW elr r rΨ = Ψ + Ψ for steric interactions Ψel is replaced by Ψsteric

Hard Sphere Mapping

L

a a

Steric stabilization Electrostatic stabilization "Charged spheres"

a aeff

Ψ / kT

r

≈1 3(1 )effLa

φ φ= +

behavior of repulsive sphere dispersions corresponds to that of hard spheres with φeff effective volume fraction increases with increasing range of interaction

Charged Spheres – Viscosity & Ionic Strength

0,1 0,2 0,3 0,41

10

100

η 0, r

φ

PS200 mM [KCl] φmax,exp

10 0.47 1 0.37 0.1 0.33

Horn, Bergenholtz, Richtering, Wagner, Willenbacher, J Coll Interface Sci 2000

0,0 0,2 0,4 0,6 0,8 1,01

10

100PS310

10 mM [KCl] 1 0.1

PS200 10 mM [KCl] 1 0.1

PS120 50 mM [KCl] 10 1 0.1

Quemada K-D

φ / φmax,exp

η 0,r

max

max, expeff

φφ φφ

=

gel-like, crystalline

l i q u i d

Viscosity and Particle Volume Fraction

10-2 10-1 100 101 102 10310-2

10-1

100

101

102

103

styrene / acrylatedispersion

up down φ = 0.49 φ = 0.47 φ = 0.45 φ = 0.44 φ = 0.43 φ = 0.42 φ = 0.40 φ = 0.35

η / Pas

γ / s-1.

two phase, weak attraction ?

10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 -2

10 -1

10 0

10 1

10 2

φ = 0.48

radius 35 nm 45 nm 65 nm 95 nm 125 nm

η / Pa s

γ / s -1

Effect of Particle Size on Viscosity

.

at high shear rates hydrodynamic forces dominate over colloidal forces → η independent of radius a

η increases with decreasing particle radius a, since φeff increases at constant φ

Viscosity of Bimodal Dispersions - 1

η 0,r

ξs

10 3

10 1 1.0 0.8 0.4 0 0.6 0.2

10 2

Chong et al. 1971

size ratio

30 µm < d < 230 µm

φ = 0.6

φ = 0.65

7.3

22.2

2.7

7.3

glass beads in PIB

ξ s 1.0 0 0.2 0.4 0.6 0.8

10 3

10 1

10 2

Rodriguez et al. 1992

φ = 0.56

φ = 0.58

PMMA in bromoforme

size ratio: 141 nm 84 nm

= 1.7

η 0,r

Non-Brownian Hard Spheres Colloidal Hard Spheres

without colloidal interactions

viscosity minimum for

• small particle fraction ξs = 25-30%

• size ratio σ as large as possible when colloidal interactions get relevant

φ → φeff

increase in φeff more pronounced

for small particles

→ optimum size ratio σ

Viscosity of Bimodal Dispersions - 2

viscosity

size ratio 10 1 5

without colloidal interactions

with colloidal interactions

Dames & Willenbacher, Rheol Acta, 2001

Viscosity of Bimodal Dispersions - 3

σ = 4.3

σ = 2

Willenbacher et al., Adhesives & Sealants, 2003

aqueous polymer dispersion φ = 0.62

Shear Thickening Occurs in Suspensions of...

Non-Brownian particles quartz PVC CaCO3

clay glass beads iron pigments starch blood cells

Colloidal particles polymer silica (SiO2) ceramics (Al2O3) iron oxide

Ribcap® New Soft Helmet Turns Hard in Crash

Origin of Shear Thickening Shear thickening results from the flow-induced formation of transient particle clusters

Viscosity increase because of the anisotropic shape of the clusters and the increased effective particle volume fraction due to trapped solvent

Cluster formation controlled by the balance of hydrodynamic force needed to push particles together and the repulsive colloidal (often also called thermodynamic) forces

so-called "Hydrocluster"

shear rate / s-1

viscosity / Pa s; turbidity / %

viscosity

turbidity

Bender+Wagner, J Rheo 1996

silica particles in tetrahydrofurfural alcohol (index-matched)

φ=0.65

strong increase in turbidity supports cluster formation hypothesis

Shear Thickening & Particle Interaction

electrosteric repulsion increases with increasing pH ⇒ shear thickening shifted to higher stresses and viscosity increase less pronounced ⇒ low shear viscosity increases strongly

Laun, Ang. Makromol Chem 1984

σ

Attractive Particle Interactions - Outline

• Structure of suspensions containing attractive particles

• Mechanisms of aggregation / flocculation

• Rheological features of weakly and strongly flocculated dispersions

yield stress and storage modulus

• Viscosity reduction due to weak attractive interactions

• Capillary forces in suspensions

Structure of Attractive Particle Suspensions

Weitz & Huang 1984

fractal aggregate structure

coagulation into compact solid aggregates and phase separation into solid and liquid fraction not considered here !

• flocs immobilize water, feff > f ® strong shear thinning

• above fc flocs form sample spanning network, percolation ® elastic, gel-like behaviour G¢ > G², yield stress

• shear-induced break-down and recovery of floc structure ® thixotropy

Flocculation of Charged Particles

24

6 349.6 tanh

4sB

critH Bb

ezk TnA k Tz l

ψ → =

2

0458 0,7

br B

r

elk T

nm nm

πε ε

ε

=

≈ ≈

( )

1 4 / 3

2 / 30

0.36 ε ε

−

→ = b scrit

r H

l QnA

changing ionic strength by adding salt changing surface charge by varying pH (or other physico-chemical parameters)

calculate critical ion concentration from DLVO-theory

Ymax = 0 and Y¢ = 0 at Ymax = 0

with Bjerrum length

61

critnz

Schulze-Hardy rule, effectiveness of multivalent ions !

weak surface potential, symmetric electrolytes

in water at room temperature

21

z

Flocculation of Sterically Stabilized Particles

stability criterion 12

− Ψ < → ≈ HvdW crit

aAkT hkT

sterically stabilizing layer must prevent particles surfaces to come closer than hcrit

1 2.52 10

≥ ≈ → ≈HPoly crit Poly

aAL h with LkT

LPoly varies strongly with temperature, especially around the q-temperature

good solvent = repulsive steric force poor solvent = attractive steric force

ΨvdW

r-kT

hcrit

Lpoly

a

hcrit

Flocculation by Addition of Polymers • depletion flocculation

• bridging flocculation dissolved polymer molecules attach to at least two particles requires affinity of polymer to particle surface long polymer chains needed in order to reduce loss of entropy

a

asmall center of polymer coil (or small particle) can not enter the shaded area = "excluded volume" ® osmotic pressure pushing large particles together reduction of excluded volume – entropic phenomenon !

non-adsorbing polymer needed attraction strength ~ polymer concentration attraction range ~ volume of polymer chain

Rheology of Flocculated Suspensions

yγσ φ weak and strong flocculation γ » 3

computer simulation chemically limited aggregation γ = 4.4 diffusion limited aggregation γ = 3.5

G¢ independent of frequency and G¢ >> G²

Yield stress

Storage Modulus

y aδσ strong flocculation δ » -2

' G αφ

' G aβ

strong flocculation α » 2.5 - 5

strong flocculation β » 0 weak flocculation β < 0

Yield Stress of Flocculated Systems

∼ φ3

Buscall et al. 1998

polystyrene particles a = 245, 480 and 1700 nm in water flocculated by adding BaCl2

3

2φσ y a

Larson S. 347 Abb. 7.18

Leong et al., Trans Royal Soc Chem 1993

Flocculation Induced by pH

"blocky"-shaped ZrO2 particles a = 150 nm in water

φ = 0.242 φ = 0.213 φ = 0.184 φ = 0.145 φ = 0.124

surface charge changes with pH strong vdW attraction at isoeelectric point

i.e.p.

317.28 K

306.20 K 304.17 K 303.16 K

302.16 K

308.13 K

/ s-1

Larson S. 339 Abb. 7.8

Woutersen & de Kruif , J. Chem. Phys. 1991

Viscosity of Flocculated Systems

octadecyl grafted SiO2 particles in benzene φ = 0.367

Thermoreversible gelation of sterically stabilized suspensions

ηr

0.85 % 1.0 %

0.6 %

0.5 % 0.4 %

0.1 %

Depletion Flocculation

ηr

acrylate particles a = 157 nm φ= 0.4 in "white spirit" with polyisobutene Mw = 411.000 g/mole

Buscall et al., J Rheo, 1993