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8/3/12 1 Suspension Rheology Jan Mewis Professor Emeritus Department of Chemical Engineering K. U. Leuven Belgium Norman Wagner Professor Department of Chemical Engineering University of Delaware USA 1.Introduction: Course Overview 2 Objectives The course is designed to be an introduction to the rheology of colloidal dispersions with emphasis on practical measurement and interpretation of rheological measurements on colloidal dispersions. The object is to provide the participants with: Qualitative understanding of the various phenomena that contribute to the rheology of suspensions; Scaling relations and quantitative laws to predict the basic rheology of such systems; Strategies to measure, characterize and design suspensions with well defined processing or application properties.
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Page 1: Suspension Rheology

8/3/12

1

Suspension Rheology

Jan Mewis Professor Emeritus

Department of Chemical Engineering K. U. Leuven

Belgium

Norman Wagner Professor

Department of Chemical Engineering University of Delaware

USA

1. Introduction:

Course Overview

2

Objectives

The course is designed to be an introduction to the rheology of colloidal dispersions with emphasis on practical measurement and interpretation of rheological measurements on colloidal dispersions. The object is to provide the participants with:

–  Qualitative understanding of the various phenomena that contribute to the rheology of suspensions;

–  Scaling relations and quantitative laws to predict the basic rheology of such systems;

–  Strategies to measure, characterize and design suspensions with well defined processing or application properties.

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8/3/12

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3

Outline Day 1

1)  Rheological Concepts and Rheological Phenomena in Colloidal Dispersions (1.5 hrs)

a)  Basic rheological concepts b)  Introduction to Colloids c)  Overview of rheological phenomena in suspensions (based on case studies)

2)  Hydrodynamic Effects (Suspensions of Large Particles) (1 hr) a)  Dilute systems: Relative viscosity and Einstein relation b)  Semi-dilute systems: Batchelor relation c)  Concentrated systems (maximum packing, viscosity-concentration relations,

effect of particle size distribution) 3)  Suspensions of Brownian Particles (1.5 hrs)

a)  Mechanism of Brownian motion b)  Contribution of Brownian motion to the viscosity c)  Viscoelasticity in suspensions of Brownian Hard Spheres, scaling relations

4)  Colloidally Stable Suspensions (2 hrs) a)  Electrostatic and Steric stabilization, resulting suspension structure b)  Effect of interparticle repulsion on dilute suspensions c)  Viscosity of concentrated stable suspensions, scaling relations d)  Viscoelastic effects, link to interparticle potential, scaling relations e)  Shear thickening

4

Outline Day 2

5)  Flocculated Suspensions (2 hrs) a)  Mechanisms of flocculation (electrostatic, depletion, bridging…) b)  Structure of flocculated systems (flocs, agglomerates, particle gels; their

description in RDF, fractals, percolation theory, stat diagrams). c)  Viscosity of dilute, flocculated systems d)  Gels and glasses e)  Thixotropy (reversible time effects)

6)  Rheological Measurements of Suspensions (1.5 hrs) a)  Special requirements and problems (based on case studies) b)  Measurement strategies

7)  Case studies(1.5 hrs) a)  Flocculation vs. bridging b)  Electrosteric stabilization

8)  Advanced Topics in Colloidal Suspension Rheology (2-3 hrs)

a)  Suspension in viscoelastic media (filled polymers, nanocomposites) b)  Suspensions containing non-spherical particles (fibers, rods, ...)

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Instructor Introduction

•  With over 50 years of academic and industrial research experience in the colloid rheology laboratory, including over 100 publications and patents on the topic, Professors Mewis and Wagner have prepared a short course designed for introducing a beginning colloid rheologist to the field. This course is based on a text currently in progress. Both have lectured extensively on the topic and have taught short courses at both the beginner and more advanced levels, including courses for the Society of Rheology in the US and the European Rheology Society in Europe.

•  Web Cites with Detailed Instructor Information: •  Mewis: www.kuleuven.be/cv/u0010977e.htm •  Wagner: www.che.udel.edu/wagner

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1

Outline Day 1 1)  Rheological Concepts and Rheological Phenomena in Colloidal

Dispersions (1.5 hrs) a)  Basic rheological concepts b)  Introduction to Colloids c)  Overview of rheological phenomena in suspensions (based on case studies)

2)  Hydrodynamic Effects (Suspensions of Large Particles) (1 hrs) a)  Dilute systems: Relative viscosity and Einstein relation b)  Semi-dilute systems: Batchelor relation c)  Concentrated systems (maximum packing, viscosity-concentration relations,

effect of particle size distribution) 3)  Suspensions of Brownian Particles (1.5 hrs)

a)  Mechanism of Brownian motion b)  Contribution of Brownian motion to the viscosity c)  Viscoelasticity in suspensions of Brownian Hard Spheres (scaling relations)

4)  Colloidally Stable Suspensions (2 hrs) a)  Electrostatic and Steric stabilization, resulting suspension structure b)  Effect of interparticle repulsion on dilute suspensions c)  Viscosity of concentrated stable suspensions, scaling relations d)  Viscoelastic effects, link to interparticle potential, scaling relations e)  Shear thickening

Suspension Rheology

Jan Mewis Professor Emeritus

Department of Chemical Engineering K. U. Leuven

Belgium

Norman Wagner Professor

Department of Chemical Engineering University of Delaware

USA

1. Introduction:

a. Rheological Concepts

Page 5: Suspension Rheology

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3

Goals: -  define basic rheological concepts

-  present basic constitutive equations

-  introduce viscoelasticity

-  provide an overview of measurement geometries

Rheological Concepts

4

Rheological Concepts

Rheology is:

The Science of Deformation and Flow of Matter

Fluid Mechanics should provide:

•  A description of the flow field (history): kinematics

•  A description of the stress distribution: dynamics

This requires:

•  An intrinsic relation between kinematics and dynamics for a given material or rheological constitutive equation

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Rheological Concepts Kinematics: Shear Flow

Shear Rate = Relative motion of layers

between sliding parallel plates:

Shear rate = dvx/dy = V/d =

6

Rheological Concepts

Dynamics

Shear stresses:

Normal stresses:

Normal stress differences:

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Overview

- Stress = Force/area (σxy: Pa)

- Shear rate = velocity gradient (dvx/dy= :s-1)

- Viscosity η= ratio of the two ( : Pa.s)

- Newtonian fluid: the ratio η is constant

Rheological Concepts

8

Rheological Concepts

Kinematics: Elongational Flow

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Liquid η [Pa.s]

Molten glass (T~500°C) 1012

Asphalt 108 Molten Polymers 103-105

Syrup 101

Paints 100-101 Glycerin 100

Light Oil 10-1 Water 10-3 Air and other Gasses 10-5

Rheological Concepts

(after Macosko, VCH 1994)

Viscosities

10

Process shear rate [s-1]

Sedimentation 10-6-10-4

Leveling 10-3-100 Extrusion 100-102

Chewing 101 -102

Mixing 101-103 Spraying, brushing 103 -104

Rubbing 104 -105 Injection molding 102 -105 Coating flows 104 -106

Rheological Concepts

Shear rates

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2 parameters

n<1 : shear-thinning n>1: shear-thickening

Rheological Models

Power Law Model

12

Rheological Models

Cross Model 4 parameters

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13

Bingham

Yield stress (Bingham)

Rheological Models

Yield Stress Models

Herschel-Bulkley

14

Viscoelastic Behavior

Oscillatory Motion

Strain:

Stress components:

Viscoelastic response:

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Viscoelastic Behavior

Simple Models

Spring Dashpot

Maxwell Kelvin-Voigt

Maxwell:

16

Viscoelastic Behavior

Oscillatory flow of a Maxwell fluid

Generalized Maxwell model:

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Rheological Measurements

Drag Flows: Rotational devices

Couette Cone-and-plate Parallel plates

(+ pressure driven devices: capillary)

18

Rheological Concepts

References for further reading

• C.M. Macosko, Rheology, Principles, Measurements and Applications, VCH 1994

• F.A. Morrison, Understanding Rheology, Oxford U.Pr., 2001

• R. Tanner, Engineering Rheology, Oxford U.Pr.. 1988

• R.G. Larson, The structure and rheology of complex fluids, Oxford U.P. 1988

• J. Mewis and N.J. Wagner, Colloidal Suspension Rheology, Cambridge U.P. 2011

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Suspension Rheology

Jan Mewis Professor Emeritus

Department of Chemical Engineering K. U. Leuven

Belgium

Norman Wagner Professor

Department of Chemical Engineering University of Delaware

USA

1. Introduction:

b. Introduction to Colloids

2

Goals

•  The goals of this section are to: –  define the basic length & time scales in the

colloidal domain – define the forces and energies acting on

single particles – define Brownian motion –  illustrate the most common interparticle

interactions – discuss collective phase behavior

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Topics

•  Colloidal sizes, energies, forces •  Single particle properties

–  Brownian motion –  Sedimentation –  Intrinsic viscosity –  Electrophoretic mobility

•  Colloidal Interactions –  Dispersion forces –  Electrostatic repulsion –  Polymers –  Hydrodynamic interactions

•  Collective Behavior

4

Types of Colloids

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Types of Colloids Plate-like,

oblate Irregular,

faceted

Oblate SiO2 SiC whiskers

Precipitated CaCO3

Al2O3 (Gibbsite) Precipitated CaCO3

Kaolin

Fe203(0H) (Goethite)

Fumed silica

Acicular, prolate

Lead sulfide

6

Types of Colloids Plate-like,

oblate Irregular,

faceted

Oblate SiO2 SiC whiskers

Precipitated CaCO3

Al2O3 (Gibbsite) Precipitated CaCO3

Kaolin

Fe203(0H) (Goethite)

Fumed silica

Acicular, prolate

Lead sulfide

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•  ~1 nm ~ 1 micron •  Brownian Motion •  Surface Forces Dominate

C=O O

Na +

a

The Colloidal World* “FLATLAND”, Abbott….

r 2a

Ψ 1/κ

ρ = ρ0 exp −Ψ( )∇2Ψ = −ρ

8

Colloidal Surface Effects

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Why 1 nm to 1 micron

•  >~1 nm: Large enough that solvent can be considered a continuum

•  <~1 micron: Small enough that Brownian motion is still an important force….

10

Brownian Motion

http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/brownian.html

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Colloidal Energy Scale

•  The fundamental unit of energy in the colloidal word is the thermal energy, kT

•  For gases, the translational energy of an ideal gas of hard spheres is 3/2 kT

•  k=Boltzmann’s constant, 1.381e-23 J/K •  At 298 K, the energy is kT=4.1e-21 J (or ~ 4 zeptojoules!)

12

Colloidal Forces

•  Forces scale on kT/a, which for a 1 micron radius particle is 4.1x10-15 N, or 4.1 fN (femtoNewtons)

•  Gravity is g*mass, the mass of a colloid is ~4*(1x10-6)3/1000=4x10-15kg, which yields a force of about 40 fN. In solution, this can be nearly eliminated by density matching.

•  Droplet forces scale on the surface tension times the size, which for water drops of 1 micron would be ~72 mN/m*1x10-6m~7x10-8 N.

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Colloidal Stresses

•  Remember Energy scales on kT, which is about 4.1x10-21 J.

•  Stress is Force/area or Energy/volume, the volume of a 1 micron colloid is

~π(1x10-6)3/6=5x10-19m3 which yields a characteristic stress of

(Note, this increases by 106 for 10nm Particles)

14

Brownian Motion

•  Stokes drag, F=f*U, f=6πµa •  Thermal force is kT/a, and the distance

over which is applies is a •  Particle flux=U*C~DC/a, Thus, we find the

Stokes-Einstein- relationship:

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Stokes-Einstein-Sutherland (a small correction of historical

interest) http://www.physik.uni-augsburg.de/theo1/hanggi/History/Robert_Brown.pdf

Stokes-Einstein-Sutherland equation? As a historical side note, Einstein did not, in fact, have the precedence on the result above. Earlier in 1905 (March), a Scotsman/Australian named William Sutherland published a very similar derivation of the Stokes-Einstein equation (which he had publicly presented in a conference already in 1904). It is not known, why the Stokes-Einstein equation is not known today as the Stokes-Einstein-Sutherland equation instead (although some authors have recently suggested it).

William Sutherland (1859-1911) Sutherland’s article was published in the Philosophical Magazine, a prestigious and well known journal. In addition, in 1905 he was already quite famous. For example, he was one of the two people outside Europe (the other one was J. Willard Gibbs) who were invited to a conference held in honor of Ludwig Boltzmann in 1906. (Einstein was not). You can get Sutherland’s paper, in addition to other Brownian-motion-related articles, from Peter Hänggi’s web page: http://www.physik.uni-augsburg.de/theo1/hanggi/History/BM-History.html

16

Brownian Motion*

* Robert Brown, Botanist, 19th century.

µ = 10-3 Pa-s

T = 300 K

k = 1.38x10-23J/K

a = 1x10-6 m

THUS:

Do = 2x10-13 m2/s

or, if a=1x10-8m,

Then:

Do = 2x10-11m2/s

solvent

Colloid

Brownian Relaxation Time:

τ=a2/Do , 5 microseconds – 5 seconds

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Perrin’s data

18

Diffusion

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Sedimentation

buoyancy

gravity

friction

For spheres, f=6πμa, and hence:

20

Sedimentation & Diffusion

One can determine the colloidal particle’s mass by a combined measurement of sedimentation and diffusion, as:

S=sedimentation coefficient

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Sedimentation Equilibrium

Balance the diffusive flux, given by Fick’s law, and the sedimentation, assuming no phase transitions, one finds… (barometric formula)

x

22

Dilution Viscometry

•  [η] is the intrinsic viscosity, in units cc/g This gives the hydrodynamic volume of a colloid. [η]=2.5/ρ for Hard Spheres

•  k is the Huggins Coefficient, it characterizes the excluded volume k=1 for hard spheres

“The Huggins Coefficient for the Square-Well Fluid”, J. Bergenholtz and N. J. Wagner, I & EC Research, 33, 2391-2403, 1994.

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•  q~1µC/cm2 •  Ψ~kT/e=25.7mV (100 is high, <25 is low) •  Solve Poisson Boltzmann Equation for

potential around a sphere

C=O O

Na +

a

Electrostatic Potentials

r 2a

Ψ 1/κ

ρ = ρ0 exp −Ψ( )∇2Ψ = −ρ

24

Electric Double Layer

counterions

Stern Layer

Plane of shear at surface plus adsorbed counterions

ζ-potential “zeta” is associated with the potential at this plane

ζ is determined from electrophoretic mobility

Potential has a

capacitance part in the adsorbed

layer, followed by the diffuse

layer

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25

Electrostatic Screening Length “Debye Length” κ-1

Example calculation

0.01M 1:1 electrolyte

Water (80 ε0) at 298 K

κ-1=3 nm

≈ CS-1/2

≈ zi-1

26

Salt effects on Debye Screening Length

Molarity Z+:Z- κ-1 (nm)

0.001 1:1 9.61 2:2 4.81 3:3 3.20

0.01 1:1 3.04 2:2 1.52 3:3 1.01

0.1 1:1 0.961 2:2 0.481 3:3 0.32

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pH effects on surface charge: Silica

Surface charge (as number of OH- groups per square nanometer) for silica (Ludox) in water showing the effect of pH and added electrolyte (NaCl, Molar). (Adapted from The Chemistry of Silica, by R.K. Iler, John Wiley & Sons, New York, 1979.)

28

Electrostatic Stabilization

Charge on Particles leads to formation of double layer in solution.

Overlap of the electrical double layer leads to a repulsive force, which is primarily osmotic in source.

The charge can arise from dissociated charge groups or adsorbed ions (clays) or surfactants.

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Electrostatic Stabilization

h

30

Dispersion Forces:

Instantaneous fluctuating dipole

Induces a dipole in the neighboring particles

This force is ubiquitous and depends on the difference in dielectric constant between the particles and the solvent.

h

A= Hamaker constant

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Connection to Work of Adhesion

Separation h=h0 separation h>h0

Equate the work done to the vdW dispersion potential for flat plates…

32

van der Waals forces depend on solvent

2 2

1

Typical values

A (PS-water) 1.4e-20 J

A (PS-vacuum) 6.5e-20 J

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33

Dispersion Forces Depend on Geometry

34

Derjaguin Landau Vervey Overbeek (DLVO) Hypothesis

A Russian group Derjaguin, Landau, and others. and the Dutch group, were both working on the same problems, between 1939 and 1945. WWII prevented them from knowing of each other’s work. Derjaguin and Landau published their important paper in 1941. Verwey published a few preliminary papers, some in the Dutch language, in 1940, 1942, 1944 and 1945 and Verwey and Overbeek prepared their monograph in that period.

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DLVO Potential

Electrostatic Repulsion

van der Waals attraction

secondary minimum

primary minimum

Born Repulsion

36

Electrostatic Interactions: Effects of Salt

Critical Coagulation Concentration CCC “Electrolyte-Induced Aggregation of Acrylic Latex I. Dilute Particle

Concentrations” Leo H. Hanus, Robert U. Hartzler and Norman J. Wagner, Langmuir, 17(11), 3136-3147, 2001.

Page 31: Suspension Rheology

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Critical Coagulation Concentration and Hamaker

Constant Measurements

Units, ions per m3

38

POLYMER & COLLOIDS

Polymer-Colloid Mixtures: the possible effects of added polymer of interest to colloid stability

From: Hiemenz and Rajagopalan, “Principles of Colloid and Surface Chemistry”, 3rd ed. Marcel Dekker, Inc. New York, 1997.

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The effect of polymer depends on:

• Adsorption

• Concentration

• Solvent Quality

Polymers & Colloids

From: Hiemenz and Rajagopalan, “Principles of Colloid and Surface Chemistry”, 3rd ed. Marcel Dekker, Inc. New York, 1997.

40

The stabilization of grafted or adsorbed polymers is characterized by the volume of overlap times the increase in osmotic pressure due to the increase in polymer segment density in the overlap region

Polymer Depletion From: Hiemenz and Rajagopalan, “Principles of Colloid and Surface Chemistry”, 3rd ed. Marcel Dekker, Inc. New York, 1997.

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Polymer Depletion induced Attraction is calculated from the volume of overlap of the excluded layers time the LOSS in osmotic pressure due to the polymer exclusion!

Polymer Depletion From: Hiemenz and Rajagopalan, “Principles of Colloid and Surface Chemistry”, 3rd ed. Marcel Dekker, Inc. New York, 1997.

42

Depletion Potential

Depletion potential shown for varying polymer concentration and polymer molecular weight.

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Phase Diagram for HS Colloidal Dispersions

Cheng Z, Chaikin PM, Russel WB, Meyer WV, Zhu J, Rogers

RB, Ottewill RH MATERIALS & DESIGN 22 (7):

529-534 OCT 2001

Pusey, P. N. & van Megen, W. Nature 320, 340–342 (1986).

44

HS Phase Diagram

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45 Bergenholtz & Wagner, PhD Thesis, UofD,1995

Bergenholtz and Fuchs, Phys. Rev. E 59, 5706 (1999)

Phase diagram of a “sticky” hard sphere

46

Critical point

Phase diagram of Square well Fluid (width =50%)

(Lakshmi Krishnamurthy, PHD UD, 2005)

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47

Effects of Electrolyte on Stability

48

Summary I

•  Stokes drag 6πµa •  Brownian motion ~kT/a •  Diffusivity D~kT/ 6πµa •  Peclet Number: •  Double layer, Debye screening length

(~3nm 10mM), electrophoretic mobility •  Intrinsic viscosity [η]=2.5/ρ

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Summary II

•  Electrostatic interactions (surface charge, electrolyte)

•  Dispersion (london-van der Waals: materials, size, shape)

•  DLVO Theory, kinetic stability •  Hydrodynamic (size, shape, medium viscosity) •  Polymer (adsorbed vs. free, steric vs. depletion,

solvent quality, graft density) •  Phase behavior

50

References for further reading

•  Hiemenz and Rajagopalan, “Principles of Colloid and Surface Chemistry”, 3rd ed. Marcel Dekker, 1997

•  Russel, Saville, and Schowalter, “Colloidal Dispersions”, Cambridge University Press, (available in paperback), 1989

•  Hunter, “Foundations of Colloid Science, Vol. I & II”, Oxford Science Publications, Clarendon Press, Oxford, 1989

•  Jaycock, M.J. and Parfitt, G.D., “Chemistry of Interfaces”, Ellis Horwood, 1981

•  Israelachvili, J., “Intermolecular and Surface Forces”, 2nd ed. Academic Press, 1992

•  Evans, D.F. and Wennerström, H., “The Colloidal Domain”, VCH, 1994

•  Larson, R.G. “The structure and rheology of complex fluids”, Oxford, 1999

Page 38: Suspension Rheology

1

Suspension Rheology

Jan Mewis Professor Emeritus

Department of Chemical Engineering K. U. Leuven

Belgium

Norman Wagner Professor

Department of Chemical Engineering University of Delaware

USA

1. Introduction:

c. Overview of Rheological Phenomena

2

Outline Day 1 1)  Rheological Concepts and Rheological Phenomena in Colloidal

Dispersions (1.5 hrs) a)  Basic rheological concepts b)  Introduction to Colloids c)  Overview of rheological phenomena in suspensions (based on case studies)

2)  Hydrodynamic Effects (Suspensions of Large Particles) (1 hrs) a)  Dilute systems: Relative viscosity and Einstein relation b)  Semi-dilute systems: Batchelor relation c)  Concentrated systems (maximum packing, viscosity-concentration relations,

effect of particle size distribution) 3)  Suspensions of Brownian Particles (1.5 hrs)

a)  Mechanism of Brownian motion b)  Contribution of Brownian motion to the viscosity c)  Viscoelasticity in suspensions of Brownian Hard Spheres (scaling relations)

4)  Colloidally Stable Suspensions (2 hrs) a)  Electrostatic and Steric stabilization, resulting suspension structure b)  Effect of interparticle repulsion on dilute suspensions c)  Viscosity of concentrated stable suspensions, scaling relations d)  Viscoelastic effects, link to interparticle potential, scaling relations e)  Shear thickening

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3

Rheological Phenomena

(after Lewis and Nielsen, 1968)

Viscosity: effect of volume fraction and particle size

Rel

ativ

e vi

scos

ity

Volume fraction

(section 2)

4

Rheological Phenomena

Viscosity: effect of colloidal phenomena

(section 3)

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5 Data from Laun (1984)

Rheological Phenomena

Viscosity: non-Newtonian behavior

(section 3)

6

Shear rheology of colloidal dispersions

fc= 0.51 (silica/PEG200

dispersions)

1 mm

Rheological Phenomena

Viscosity: non-Newtonian behavior, shear thickening

(section 4) (data from R. Egres, PhD, U of D)

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7 Effect of salt concentration

PS spheres, a=110 nm F = 0.40 --- hard spheres Data from Krieger and Eguiluz, ‘72

Rheological Phenomena

Viscosity: effect of repulsion forces

(section 4)

8

PMMA/PHSA suspensions - decalin/PDL15 (data of G. Ourieva, 1999)

Effect of interparticle attraction

Rheological Phenomena

(section 5)

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After (Shikata & Pearson, JoR, 1994)

Rheological Phenomena

Viscoelasticity from colloidal forces

(slide 8 from section 3) (section 3)

10 Frith et al., JCIS, 1980

Rheological Phenomena

Viscoelasticity: S/L transition and nonlinearity

(section 4)

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Time effects: Thixotropy

(G. Biebaut, 2002)

Rheological Phenomena

(section 5)

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1

Suspension Rheology

Jan Mewis Professor Emeritus

Department of Chemical Engineering K. U. Leuven

Belgium

Norman Wagner Professor

Department of Chemical Engineering University of Delaware

USA

2. Hydrodynamic Effects

2

Outline Day 1 1)  Rheological Concepts and Rheological Phenomena in Colloidal

Dispersions (1.5 hrs) a)  Basic rheological concepts b)  Introduction to Colloids c)  Overview of rheological phenomena in suspensions (based on case studies)

2)  Hydrodynamic Effects (Suspensions of Large Particles) (1 hrs) a)  Dilute systems: Relative viscosity and Einstein relation b)  Semi-dilute systems: Batchelor relation c)  Concentrated systems (maximum packing, viscosity-concentration relations,

effect of particle size distribution) 3)  Suspensions of Brownian Particles (1.5 hrs)

a)  Mechanism of Brownian motion b)  Contribution of Brownian motion to the viscosity c)  Viscoelasticity in suspensions of Brownian Hard Spheres (scaling relations)

4)  Colloidally Stable Suspensions (2 hrs) a)  Electrostatic and Steric stabilization, resulting suspension structure b)  Effect of interparticle repulsion on dilute suspensions c)  Viscosity of concentrated stable suspensions, scaling relations d)  Viscoelastic effects, link to interparticle potential, scaling relations e)  Shear thickening

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3

Overview

§  Experimental observations

§  Flow fields in suspensions

§  Viscosity of dilute systems: Einstein equation

§ Viscosity of semi-dilute systems: Batchelor equation

§  Viscosity of concentrated systems

§  Particle size distribution

§ Complexities (anisotropy, normal forces and migration)

4

Experimental Observations

(after Lewis and Nielsen, 1968)

Viscosity: effect of volume fraction and particle size

Mooney equation:

Rel

ativ

e vi

scos

ity

Volume fraction

Relative viscosity:

Page 46: Suspension Rheology

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5 After Chong et al., 1971

Experimental Observations

Viscosity: effect particle size distribution

6

Viscosity: maximum packing

(Chong et al., 1971)

Experimental Observations

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7

Experimental Observations

(Zarraga et al., 2000)

Viscosity: apparent shear rate dependence

8

Flow around a sphere

1) Effect of flow on particle: translation and rotation

2) Effect of particle on flow: extra energy loss (deviation of stream lines, friction on surface) meaning a higher effective viscosity

Viscosity increase with volume fraction:

(Einstein, 1906, 1911)

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9

Assumptions of Einstein equation

1. Suspending medium is a continuum, it is Newtonian and incompressible 2. Creeping flow 3. No sedimentation 4. No slip at particle surface 5. Velocity profile is not altered by the presence of particles:

6. Particles much smaller than gap:

7. The suspension is sufficiently dilute that the particles never interact with each other

10

Closed Stream Lines

Compression

Elongation

Open Stream Lines

Closed Stream Lines

Open Stream Lines

Particle Trajectories

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Semi-dilute Suspensions •  Semi-dilute: only pairwise interaction (no multiple ones).

•  Pairwise collisions increase the viscosity .

•  Calculation of requires including the distribution of all possible interactions, which depends on structure and type of flow.

•  For shear flow assumptions have to be made about initial structure because the closed trajectories (e.g. random structure).

Viscosity equation:

With, for elongational flow: c2 = 7.6

for shear flow: c2 = 5.0 (Batchelor and Green, JFM 56,401, 1972, Wagner and Woutersen, JFM 278,267,1994)

12

Consider the concentrated suspension as being particles immersed in a

fluid comprised of the other particles, thus: accessible volume, where k is a packing parameter.

Concentrated Suspensions

•  Higher order particle interactions occur and give rise to higher order terms in .

•  There is a maximum packing for the particles, beyond which no flow is possible without dilation; there the viscosity diverges.

•  Only semi-empirical expressions for the viscosity exist.

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13

integrating

Thus:

or

With limits

yields

(Krieger and Dougherty, 1972)

Concentrated Suspensions

14

Viscosity (hydrodynamic effects)

Describing the dependence on volume fraction

(Sierou and Brady, 2002)

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15

Shapiro & Probstein, PRL 1992

25% small

Viscosity (hydrodynamic effects)

The effect of particle size distribution

16

Mixing particle sizes

(after Chong et al., 1971)

Effect of size ratio and total volume fraction

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17

(Parsi and Gadala-Maria, 1987)

Complexities

Flow-induced anisotropy

(Gadala-Maria and Acrivos, 1980)

angular dependence of the radial distribution function)

flow reversal

18

Complexities Normal stress differences

Ø  Flow-induced anisotropy causes normal stress differences in suspensions Ø  Both N1 and N2 are negative, they are small and scale with

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19

Complexities

Migration/Diffusion

(Gadala-Maria and Acrivos, 1980) (after Ovarlez et al., 2006)

Ø  self diffusion Ø  gradient diffusion

20

Complexities

Migration/Diffusion

(after Ovarlez et al. (2006)

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21

Non-Brownian Spheres

References for further reading

•  van de Ven T.G.M., Colloidal Hydrodynamics, Acad. Press, London 1992 (hydrodynamics).

•  Zarraga I.E. et al., J. Rheol. 44, 185, 2000 (rheol. measurements). •  Sierou A. and Brady J.F., J. Rheol. 46, 1031, 2002 (simulation,

microstructure, rheology). •  Breedveld V. et al., J. Chem. Phys. 116, 10529, 2002 (diffusion). •  Leshansky A.M. and Brady J.F., J. Fluid Mech. 527, 141, 2005

(simulation, migration). •  Matas J.-P. et al., J. Fluid Mech. 515, 171, 2004 (inertia). •  Wagner N.J. and Woutersen A.T.J.M., J. Fluid Mech. 278, 267,

1994 (particle size distribution, dilute). •  Dames B. et al., Rheol. Acta 40, 434, 2001 (particle size

distribution). •  Mewis J and N.J. Wagner: chapter 2 in: Colloidal Suspension

Rheology, Cambridge U.P. 2011.

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Suspension Rheology

Jan Mewis Professor Emeritus

Department of Chemical Engineering K. U. Leuven

Belgium

Norman Wagner Professor

Department of Chemical Engineering University of Delaware

USA

3. Suspensions of Brownian Particles (Hard Spheres)

2

Outline Day 1

1)  Rheological Concepts and Rheological Phenomena in Colloidal Dispersions (1.5 hrs)

a)  Basic rheological concepts b)  Introduction to Colloids c)  Overview of rheological phenomena in suspensions (based on case studies)

2)  Hydrodynamic Effects (Suspensions of Large Particles) (1 hrs) a)  Dilute systems: Relative viscosity and Einstein relation b)  Semi-dilute systems: Batchelor relation c)  Concentrated systems (maximum packing, viscosity-concentration relations,

effect of particle size distribution) 3)  Suspensions of Brownian Particles (1.5 hrs)

a)  Mechanism of Brownian motion b)  Contribution of Brownian motion to the viscosity c)  Viscoelasticity in suspensions of Brownian Hard Spheres (scaling relations)

4)  Colloidally Stable Suspensions (2 hrs) a)  Electrostatic and Steric stabilization, resulting suspension structure b)  Effect of interparticle repulsion on dilute suspensions c)  Viscosity of concentrated stable suspensions, scaling relations d)  Viscoelastic effects, link to interparticle potential, scaling relations e)  Shear thickening

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3

Outline

•  Mechanisms of Brownian Motion •  Contribution of Brownian Motion to the

Viscosity •  Viscoelasticity in Suspensions of Hard

Spheres •  Scaling Relationships

4

Landmark Observations I

Laun, Die Angew. Makr. Chem. 1984

Shear viscosity depends on volume fraction and stress. Shear thinning, yielding, and shear thickening are observed

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5

Landmark Observation II

Woods and Krieger, JCIS 1970

Addition of electrolyte to an aqueous latex leads to a viscosity minimum at finite stresses. This minimum was associated with Hard Sphere behavior

6

Landmark Observations III Adapted from Russel, Saville, Schowalter, Cambridge, 1989

Relative viscosity versus dimensionless stress is independent of particle size

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7

Landmark Observations IV

Foss & Brady, J.Fluid Mech., 407, 167-200, 2000

Relative zero shear viscosity depends on volume fraction only and diverges at or near random close packing.

8

Landmark Observations V

DeKruif et al. JCP 1985

Divergence is different at high shear rates; suspensions flow under shear when they would have a yield stress at rest.

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Landmark Observations VI Shikata & Pearson, JOR, 1994.

Hard sphere suspensions show viscoelasticity, where the loss modulus is higher than the storage modulus. A characteristic relaxation time is apparent and high frequency plateau may be achieved.

10

Landmark Observations VII

N1 is relatively small, and changes sign, becoming negative at high shear rates (shear thickening).

Lee et al., JOR, 2005

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11

Brownian Hard Spheres- Review

solvent

Colloid Te

mpe

ratu

re d

oes

not m

atte

r, a

ther

mal

pot

entia

l

r

r

HS

electrostatic

steric

12

Variables: •  a particle radius •  kT thermal energy •  η medium viscosity •  Φ volume fraction •  . relaxation time •  σ shear stress •  . shear rate

Scaling

Dimensionless Groups •  Relative viscosity

•  Dimensionless stress

• Peclet Number (shear rate)

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Brownian Motion Contributes to the Stress

•  Brownian Motion acts like an effective force that keeps particles apart and sets a suspension microstructure

•  GKB, 1970

•  Force depends on g(r), the radial distribution function

•  Force per area is a stress

14

Brownian Stress cont…

•  The radial distribution function describes the location of the neighboring particles

r

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15

Brownian Stresses, cont.

•  g(2a) gives the osmotic pressure and viscosity contribution

Brady, JCP, 1993 a,b

16

Zero Shear Viscosity and Osmotic Pressure

Brady, JCP, 1993 a,b

Phan et al. PRE, 1996

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Validation by Confocal Microscopy

Direct imaging of hard sphere liquid showing: a) computer reconstruction of the measured colloidal structure at = 0.25; b) comparisons with the hard sphere radial distribution functions, and c) osmotic pressure determined from the images (from Dullens et al. PNAS, 2006).

18

Dilute Suspensions

Wagner and Woutersen, JFM, 94 Extra stress is due to Brownian motion

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19

s r

f

0.49-0.55 Freezing 0.58 Glass Transition

0.63-0.64 Random Close Packing

What is ϕmax ?

Concentrated Systems, Low Shear

20

Two “simple” questions of colloid science & rheology:

1) At what volume fraction does the zero shear viscosity diverge for Hard Sphere dispersions?

2)With what functionality does the zero shear viscosity for HS dispersions diverge?

Tem

pera

ture

doe

s no

t mat

ter,

ath

erm

al p

oten

tial

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21

Choices for

•  Random Close Packing (maximally random jammed state)

•  Well established at 0.638 •  JB micromechanical

theory suggests power law divergence with exponent of -2

•  Russel & Chaiken suggest exponential divergence

•  Ideal Glass Transition •  Experimentally ~0.58

(Pusey & van Megen) •  Power law divergence,

MCT Fuchs •  May be subtended by “hopping”, leading to exponential divergence

(Torquato, Annu. Rev. Mater. Res. 2002. 32:77–111)

22

Diverging at 0.63

Diverging at 0.58

Literature Data & Simulations

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23

Concentrated Dispersions Zero Shear Viscosity: Models

•  Krieger-Dougherty

•  Quemada

•  Brady

•  Cheng et al (Russel & Chaiken)

•  Mode Coupling Theory •  Others…..

24

SD Simulations (Brady & coworkers)

Zero shear viscosity of hard spheres calculated by Stokesian dynamics; the hydrodynamic and Brownian components are also shown separately. The inset shows the data on a semi-log plot.

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25

High Frequency Viscosity

High frequency dynamic rheology probes the hydrodynamic viscosity

26

Test of MC Theory

Fuchs et al. PRA 45, 1992

Diverges at the Glass Transition

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27

MCT captures the divergence as the glass transition is approached

Mode Coupling re-plotted

28

MCT Physical Picture: Cages

Ø Every particle is “caged” by its neighbors, each neighbor is in turn, caged by its neighbors. Ø Release from the cage of neighbors leads to flow. Ø Resistance to this release increases the viscosity. Ø Motion of one is coupled to the motion of all the neighbors, etc..

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Glass Transition

•  At the glass transition, all particles become localized in their cages and a glass is formed (with a yield stress)

•  Shearing can “melt” the glass

•  Thermally activated “hopping” (not in the theory) takes over (Doolittle equation, Eyring)

•  Hopping ceases at random close packing

30

Resolution ?

•  MCT correctly describes the powerlaw approach to the colloidal glass transition from ~20% to 2%

•  Exponential divergence may be observed due to “hopping” processes that pre-empt the glass transition

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Soft Particles “Hopping”

Slightly soft particles enable exploring how the viscosity diverges- turn over to the exponential is observed

Semi-Empirical Equation for Hard Spheres

Russel, Wagner & Mewis, JOR 2012 (submitted)

32

ηo − ′η∞

µ= 0.9φ 2 1−φ φg( )−2.46

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33

Viscoelasticity

Helgeson et al. JOR 2007 Mason and Weitz, PRL ‘95

Shikata & Pearson, JOR, 1994.

Liquid Glass

34

Glassy Dynamics

Viscoleastic measurements on polydisperse, thermosensitive dispersions of HS-like particles just below the glass transition. Lines are solutions to the MCT theory (Siebenburger et al. JOR, 2009).

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35

Modulus dependence on

Modulus scales with kT/a3

Lionberger & Russel, JOR 1994

36

Roughness leads to G’∞αω1/2

Lionberger and Russel show how particle roughness can lead to a power law behavior, often seen in real suspensions (no plateau)

Mason & Weitz, PRL95

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37

Diffusion and Viscosity – related?

Short time self-diffusion is qualitatively similar to high frequency viscosity

Stokes-Einstein-Sutherland:

38

Particle Dynamics in the Glass

Intermediate light scattering functions for sterically stabilized PMMA dispersions (a = 102 nm, 4% polydispersity) comparison with the scaling predictions of the MCT (van Megen and Underwood, PRE, 1994).

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39

Shear Melting LAOS

Increasing shear amplitude melts the glass and the sample flows

Pham et al., JOR 2008

40

Frequency Thinning

Brady, JCP, ’93. Frequency thinning theory compared to data of Van der Werff and de Kruif, JOR, ‘89

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41

Shear Thinning

deKruif et al. JCP ‘85

42

Characteristic Shear Rate for Shear Thinning: Brownian Motion*

* Robert Brown, Botanist, 19th century.

solvent

Colloid

Brownian Relaxation Time:

=a2/Do , 5 microseconds – 5 seconds

When the shear rate is faster than the Brownian relaxation time, the sample’s microstructure will be strongly affected by the shearing

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43

Shear Thinning: Master Curves

deKruif et al. JCP, 83 (1985).

Rheology of suspensions of weakly attractive particles: Approach to gelation C. J. Rueb and C. F. Zukoski, J. Rheol. 1998.

44

Shear Thinning a Consequence of Microstructure Rearrangement

Pe~0 Pe~0(1)

Fewer collisions, less stress

Guidance from Brownian Dynamics Simulations (Rastogi and Wagner, JCP 96) (i.e. no hydrodynamic interactions):

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45

Microstructure

Plots of the nonequilibrium suspension microstructure in the dilute limit for moderate to high Pe, obtained with Stokesian Dynamics. b) overlay with the sign of the Brownian force contribution to the shear stress; c) overlay with the sign of Brownian force contribution to the first normal stress difference (Foss and Brady, JFM 2000)

46

Rheological Functions

Stokesian Dynamics simulations of the relative viscosity and normalized primary and secondary normal stress differences as labeled for 45 vol% hard spheres as a function of Peclet number. Also shown is the angular average. Foss and Brady, JOR 2000.

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47

Shear Thickening: (Landmark Observations I)

Laun, Die Angew. Makr. Chem. 1984

48

Engineered Nanomaterials e.g., “house paint”

H. Martin Laun, Ang. Mak. Chem. 123/124 (1984), 335.

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49

“Jamming” picture of colloid dynamics

Nonlinear dynamics: Jamming is not just cool any more ANDREA J. LIU1 AND SIDNEY R. NAGEL2

Nature 396, 21 - 22 (05 November 1998); doi:10.1038/23819

Dispersions “jam” at high stresses

Cates et al., PRL, 1998

Colloidal Rheology Glass transition

H. Martin Laun, Ang. Mak. Chem. 123/124 (1984), 335.

50

increasing

equilibrium shear thin shear thicken

Hydrocluster Mechanism

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51

increasing

equilibrium shear thin shear thicken

Melrose, JR, J. RHEOLOGY, 48(5) 961-978 (2004) John Melrose and Robin Ball, approximate SD simulations

Hydrocluster Mechanism

52

2Ref. Brady Current Op. in Coll. & Int. Sci. 1996 Rheology & dichroism measurements1

Pe Stokesian dynamic simulations2

Mechanism of Shear Thickening

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53

•  Solids Phase: Size, Shape, Interaction, Polydispersity, Concentration •  Flow Field: Type, Time, Magnitude

•  Continuous Phase: Viscosity, Dielectric Constant, Ionic Strength

151 nm 38 nm

328 nm

f size

concentration

f=0.58

f=0.38

Shear thickens at lower shear rates for larger particles, higher concentrations

Parameters Controlling Shear Thickening

54

Mitigating Shear Thickening “The Role of Nanoscale Forces in Colloid Dispersion Rheology” NJ Wagner and JW Bender, MRS Bulletin, 2004. Krishnamurthy, Wagner & Mewis, JOR, 2005

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55

Review

•  HS dispersions show elasticity and a transition to a glass

•  HS dispersions show shear thinning and shear thickening

•  Energy scale of Brownian motion is kT •  Time scale is that of Brownian diffusion a2/

D0 •  Shear thickening occurs when shear

forces drive particles into “hydroclusters”

56

References for Further Reading

•  R.G. Larson, The Structure and Rheology of Complex Fluids (1999). •  Theory: “Microscopic theories of the rheology of stable colloidal

dispersions” Lionberger RA, Russel WB ADVANCES IN CHEMICAL PHYSICS, 111:399-474 (2000)

•  Shear Thickening: Maranzano and Wagner, JCP, JOR 2001, 2002. •  Stokesian Dynamics: Bancio and Brady, “Accelerated Stokesian

Dynamics Brownian Motion,” JOURNAL OF CHEMICAL PHYSICS 118: 10323-10332, 2003.

•  Bergenholtz et al. “The non-Newtonian rheology of dilute colloidal suspensions.” J. Fluid Mech. 278: 267-287, 2002.

•  W.B. Russel, D.A. Saville and W.R. Schowalter, Colloidal Dispersions (1989).

•  W.B. Russel, N.J. Wagner and J.M. Mewis, J. Rheology (2012) submitted.


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