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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