uva-dare (digital academic repository) rheology of dry, partially … · the de nition of rheology...

UvA-DARE is a service provided by the library of the University of Amsterdam (http://dare.uva.nl)

UvA-DARE (Digital Academic Repository)

Rheology of dry, partially saturated and wet granular materials

Pakpour, M.

Link to publication

Citation for published version (APA):Pakpour, M. (2013). Rheology of dry, partially saturated and wet granular materials.

General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s),other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).

Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, statingyour reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Askthe Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam,The Netherlands. You will be contacted as soon as possible.

Download date: 01 Jan 2021

https://dare.uva.nl/personal/pure/en/publications/rheology-of-dry-partially-saturated-and-wet-granular-materials(857a9953-7eaf-419e-acca-63975e7033e8).html

Rheology of Dry, PartiallySaturated and Wet Granular

Materials

Rheology of Dry, PartiallySaturated and Wet Granular

Materials

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad van doctoraan de Universiteit Amsterdam

op gezag van de Rector Magnificusprof. dr. D.C. van den Boom

ten overstaan van een door het college voor promotiesingestelde commissie,

in het openbaar te verdedigen in de Agnietenkapelop woensdag 27 november 2013, te 14:00 uur.

door

Maryam Pakpour

geboren te Arak, Iran

Promotiecommissie:

Promotor: Prof. dr. D. Bonn

Overige Leden: Prof. dr. B. Nienhuis

Prof. dr. P. G. Bolhuis

Dr. P. Schall

Dr. N. Shahidzadeh

Prof. dr. C. Wagner

Prof. dr. M. van Hecke

Dr. M. Habibi

Faculteit der Natuurwetenschappen, Wiskunde en Informatica.

Cover page by: Lili Farhadi

c©Copyright 2013 by Maryam Pakpour. All rights reserved.

The author can be reached at [email protected]

The research reported in this thesis has been carried out as partof the scientific program of the Netherlands Organization for Scientific Research(NWO) and the Foundation for Fundamental Research on Matter (FOM).

Contents

1 General introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Development of rheology . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Basic rheometrical concepts . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Shear stress tensor . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.2 Normal stress . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Granular materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Dry and wet granular materials . . . . . . . . . . . . . . . . . 6

1.4.2 Wet granular materials with different degrees of wetting . . . 8

1.5 Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Capillary forces between two spherical bodies . . . . . . . . . . . . . 10

1.7 Basic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.8 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.9 Non-Newtonian behaviour . . . . . . . . . . . . . . . . . . . . . . . . 14

1.9.1 Shear thinning and shear thickening . . . . . . . . . . . . . . 15

1.9.2 Yield stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.10 Rheometry; experimental techniques . . . . . . . . . . . . . . . . . . 19

1.10.1 Concentric cylinder rheometer . . . . . . . . . . . . . . . . . 19

1.10.2 Cone and plate geometry . . . . . . . . . . . . . . . . . . . . 20

1.10.3 Parallel plate geometry . . . . . . . . . . . . . . . . . . . . . 22

1.11 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

I Rheology of Dry and Partially Saturated GranularMaterials 25

2 Dissipation in wet and dry granular materials 27

2.1 Linear viscoelastic behaviour . . . . . . . . . . . . . . . . . . . . . . 27

i

ii CONTENTS

2.1.1 Oscillatory shear tests . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Flow of wet and dry granular materials . . . . . . . . . . . . . . . . . 31

2.3 Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Shear cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.2 Oscillatory shear tests . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.1 Energy for flowing granular materials . . . . . . . . . . . . . 35

2.4.2 Energy dissipated in wet and dry granular matter . . . . . . 38

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 How to construct the perfect sandcastle 43

3.1 Capillary bridges in wet granular material . . . . . . . . . . . . . . . 43

3.2 What is buckling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 Historical review . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 A mechanism for buckling . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Buckling of a column under its own weight . . . . . . . . . . . . . . 48

3.5 The shear modulus of wet granular materials . . . . . . . . . . . . . 50

3.6 Buckling of the sand columns . . . . . . . . . . . . . . . . . . . . . . 52

3.6.1 Experimental techniques . . . . . . . . . . . . . . . . . . . . 52

3.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

II Rheology of Wet Granular Materials 59

4 Flow irreversibility in granular suspensions 61

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Magnetic resonance imaging . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Flow irreversibility in granular suspensions under large amplitudeoscillatory shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.1 Experimental techniques . . . . . . . . . . . . . . . . . . . . . 63

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.1 Viscosity and particle migration . . . . . . . . . . . . . . . . 64

4.4.2 Critical strain for irreversibility . . . . . . . . . . . . . . . . . 67

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

CONTENTS iii

5 Rheology of suspensions 75

5.1 Introduction to the rheology of complex fluids . . . . . . . . . . . . 75

5.2 Different regimes of sheared granular materials . . . . . . . . . . . . 76

5.2.1 Particle-particle contacts in suspensions . . . . . . . . . . . . 77

5.3 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4.1 Shear stress behaviour and constitutive equation . . . . . . . 79

5.4.2 Viscosity of suspension . . . . . . . . . . . . . . . . . . . . . . 81

5.5 Normal stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography 85

Summary 94

Samenvatting 96

Nomenclature 98

List of publications 99

Acknowledgements 100

iv CONTENTS

1.

General introduction

1.1 Introduction

The general features of all complex materials like polymers, emulsions and suspen-sions which exhibit very complicated superimposition of liquid-like and solid-likeproperties, pushed the scientists to introduce a new way of understanding thebehaviour and properties of these materials. These complex fluids possess mechan-ical properties that are intermediate between liquid and solid behaviour. So, twocenturies after Newton and Hooke, the need for new methods to measure and char-acterize the properties of complex materials led to the creation of rheology [1–3].

Reology is the study of the relation between the applied force (outer action) anddeformation (inner reaction) of the materials which is a nonlinear behaviour forcomplex fluids. In this sense, the Newton-Stokes and Hooke Laws are the limitingcases on the border of rheology. The term “Rheology” was invented by ProfessorBingham, and it means “Science of the deformation and flow of matter”. The goalof rheological experiments is to quantify viscoelastic properties over a range of timeand deformation to relate these features to the behaviour of the material structurevia conservation equations and the mathematical models [4].

This chapter provides an overview of the basic concepts of rheology and someimportant elements of this science. We study the non-Newtonian fluids with em-phasis on granular materials and suspensions and the behaviour of these materials.Then we investigate the experimental techniques of rheometry and the effects ofthe geometry of devices to probe the mechanical responses of complex matter tounderstand the properties of these materials. The last section is devoted to thestudy of viscoelasticity and the nonlinear behaviour of the fluids with a review ofexperimental methods.

1

2 Chapter 1. General introduction

1.2 Development of rheology

In 1678 Robert Hooke in his “True Theory of Elasticity” proposed that the forcein a springy body was proportional to its extension. For a Hookean solid, “ashear stress applied to a surface results in an instantaneous deformation. Once thedeformed state is reached there is no further movement, but the deformed statepersists as long as the stress is applied” [4]. We can write this in the form:

σ = Gγ, (1.1)

where σ is the shear stress and γ and G are the strain and rigidity modulus,respectively.

Nine years after the publication of Hooke’s paper, Isaac Newton gave attention tothe problem of steady shear flow in a fluid and in his famous monograph “Principia”published in 1687, he mentioned the hypothesis associated with a steady simpleshearing flow: “the resistance which arises from the lack of slipperiness of the partsof the liquid, other things being equal, is proportional to the velocity with whichthe parts of the liquid are separated from one another”. This lack of slipperiness iswhat we now call viscosity [3,4]. According to Fig. 1.1, for a simple flow condition,when a liquid is contained between two parallel plates of which the top plate slideswith velocity U , the fluid will flow everywhere parallel to the plates. Assumingthe local velocity varies linearly across the gap, the gradient of velocity ux in they direction is given by:

duxdy

=U

h= γ. (1.2)

To generate the flow a force Fyx (in direction x) has to be applied to the upperplate and move it at velocity U . So the stress σyx is transmitted from one plateto the other and the kinematic parameter that determines the level of the internalstresses is the velocity gradient or shear rate. This relation specifies that the stressis proportional to the velocity gradient. The proportionality constant, the viscositycoefficient η expresses the resistance to flow in Newtonian fluids [5–7]:

σyx = ηduxdy

. (1.3)

Newton’s law is the simplest example of a rheological constitutive equation and theFig. 1.1 is an idealization of the flow in a typical rheometer.

In the nineteenth century, several scientists studied a wide variety of fluids andbegan to doubt the concept of liquid and solid. They carried out experimentsand found that many colloidal suspensions and polymer solutions did not obeythe simple linear relations and cannot be described by Hooke’s or Newton’s lawsalone. These “complex fluids” have the mechanical properties that are intermediatebetween ordinary solids and liquids. All these materials give viscosities that changewith increasing velocity gradient of the shear.

1.3. Basic rheometrical concepts 3

Figure 1.1: Simple shearing. Fluid is contained between two infinite parallel platesseparated by a distance d. The top plate moves with a constant velocity U in the xdirection and the lengths of the arrows between the planes show the local velocity u inthe flow.

The definition of rheology we already have would allow a study of the behaviour ofall materials, including the classical extremes of Hookean elastic solids and New-tonian viscous liquids. The first applications of rheology focused on the propertiesand behaviour of such different materials as asphalt, lubricants, paints, plasticsand rubbers. The historical perspective shows that rheological studies developedin the twentieth century and continued in the Second World War that rheologygenerated its share of research to use viscoelastic materials during the war [4].

Nowadays, the scope is even wider. Significant advances have been made in biorhe-ology and, polymer, emulsion and suspension rheology. Also there have been im-portant developments in the pharmaceutical and, food industries, and in medicalresearch. The manufacture of materials by the biotechnological route requires agood understanding of the rheology involved. All these developments and materialshelp to illustrate the substantial relevance of rheology to life following the secondhalf of the twentieth century [3, 4, 8].

1.3 Basic rheometrical concepts

Rheology studies the behaviour of various real continuous media and the relation-ship between applied forces and inner reaction (flow or deformation of the materi-als). The behaviour of a material is a relationship between forces and deformationsand a model which gives a mathematical formulation of this relationship, based onconstitutive equations. One of the goals of rheology is a search for stress versus de-formation relationships for various technological and engineering materials in orderto solve macroscopic problems related to the continuum mechanics of these mate-rials. Also this science establishes relationships between the rheological properties


Figure 1.2: Stress components in three dimensional flow.

of a material and its molecular composition content.

In this section we explain some of the fundamental concepts in rheology that de-scribe any dynamic point of a medium by these physical quantities [2].

1.3.1 Shear stress tensor

Fig. 1.2 shows a volume element of a fluid which has been considered under normaland tangential forces. For an incompressible fluid, the shear stress is defined as atangential force per unit area. When a fluid is at rest, the force causes an identicalpressure and acts as normal to the surface elements. So the stress is isotropic, andcalled the hydrostatic pressure. If the fluid is in motion, the layers of the fluidslide on each other and due to the viscosity of the fluid, the frictional force ortangential stress appears on the surface element dS. This leads to another stresstensor coefficients, σij which represents the component of the imposed stress in thei-direction on a surface with normal orientation in the j-direction (Fig. 1.2).

We can separate the parts of the stress tensor σ that corresponds to the hydrostaticpressure stresses acting in the absence of velocity gradients (for a fluid at rest inuniform translational motion). This component is diagonal and isotropic and allthree diagonal coefficients are identical. So the stress tensor could be written as:

σij = σ′ij − pδij , (1.4)

where p is the hydrostatic pressure and δij is the Kronecker delta. The other termσij is a general expression for the viscosity stress tensor. Therefore, if the forcedF is exerted on a surface dS with normal vector n (in an arbitrary direction), thecorresponding stress σ, exerted on this surface is given by:

1.3. Basic rheometrical concepts 5

Normal [Chap.

again equal to the ambient pressure. As the shear rate is increased, the normal stress differences first appear as second-order effects, so that we can write

where A, and are constants and, as implied, the normal stress differences areeven functions of the shear rate (The mathematically-minded reader may confirmthis expectation by undertalung a simple analysis for the hierarchy equations givenin Chapter 8 (eqns. which are argued to be generally valid constitutiveequations for sufficiently slow flow).

From a physical point of view, the generation of unequal normal stress compo-nents, and hence non-zero values of and arises from the fact that in a flowprocess the microstructure of the liquid becomes anisotropic. For instance, in adilute polymeric system, the chain molecules, which at rest occupy an envelopingvolume of approximately spherical shape, deform to an ellipsoidal shape in a flowfield. The molecular envelope before and during deformation is shown in Fig. 4.1.The droplets in an emulsion change shape in a similar way. In the polymeric systemat rest, forces determine the spherical shape whilst the requirement of aminimum interfacial free-energy between an emulsion droplet and the surroundingliquid ensures practically spherical droplets in the emulsion at rest. It followstherefore that restoring forces are generated in these deformed microstructures and,since the structures are anisotropic, the forces are anisotropic. The sphericalstructural units deform into ellipsoids which have their major axes tilted towards thedirection of flow. Thus the restoring force is greater in this direction than in the twoorthogonal directions. The restoring forces give rise to the normal stress componentsof eqns. 4.2. It can be appreciated, from this viewpoint of their origin, why it is thatthe largest of the three normal stress components is always observed to be a,,, thecomponent in the direction of flow. According to the principles of continuummechanics, the components can have any value, but it would be an unusualmicrostructure that gave rise to components whose relative magnitudes did not conform to a,, a,,. It is conceivable that a strongly-orientatedstructure, as found in liquid crystals, might produce such unusual behaviour incertain circumstances.

AT REST UNDER SHEAR

Fig. 4.1 The molecular envelope before and during shear deformation.Figure 1.3: The molecular cover before and during shear deformation [4].

dF

dS= [σ].n. (1.5)

The term [σ].n expresses the inner product of the second rank tensor [σ] with thevector n [6].

1.3.2 Normal stress

Normal stress differences are associated with the non-linear effects in the materialsand they did not appear explicitly in the account of linear viscoelasticity [4]. Soin the Newtonian fluids the three normal stress components are isotropic, whilefor the non-Newtonian matter, like polymeric systems, the effect of elasticity isattributed to the normal stresses generated during the shear flow. A simple shearflow profile is shown in Fig. 1.1. The flow is in the x direction, while the velocitygradient exists in the y direction. The first and second normal stress differences,N1 and N2 can be expressed in the form:

N1(γ) = σxx − σyy = γ2ψ1,

N2(γ) = σyy − σzz = γ2ψ2, (1.6)

where σii is the shear stress component in the i direction and ψ1 and ψ2 arecalled the normal stress coefficients. For Newtonian materials such as pure water,both N1 and N2 are zero in shearing flow. For non-Newtonian materials, N1 ispositive and much larger than N2 which is normally very small in magnitude andhas been reported to be negative for some materials. This suggests that elasticfluids generate extra stress above the shear stress in the direction of the flow (forcone-plate or plate-plate rheometers, this means that the surfaces of the plates orcone tend to be pushed apart). This extra stress in the elastic fluids is generateddue to stretching and alignment of the polymer chains along the streamlines.

The origin of different normal stress components or the non-zero values of N1 andN2 arises from the fact that the microstructure of a flow under shearing, becomes


anisotropic. For example, in a dilute polymeric system, the chain molecules, whichat rest occupy an approximately spherical volume, deform to an ellipsoidal shape ina flow field. Figure 1.3 shows this molecular form before and during deformation.In the polymeric system at rest, entropic forces determine the spherical shape whilein a similar way for the droplets in an emulsion, the requirement of a minimuminterfacial free energy between an emulsion droplet and the surrounding liquidensures spherical droplets in the emulsion at rest. It follows therefore that restoringforces are generated in these deformed microstructures and, since the structuresare anisotropic, the forces are anisotropic. The restoring forces give rise to thenormal stress components in Eq. 1.6. It is difficult to give reliable estimates of N2.Combined cone-plate and plate-plate force measurements, appear to give reliablevalues [9] and satisfactory results for normal stress differences. The two normalstress differences defined in this way, are characteristic of a material, and they canbe used to categorize a fluid as purely viscous or as non-linear viscoelastic [4, 10].

1.4 Granular materials

A granular material is a collection of solid particles or grains, which are in contactwith at least some of their neighbouring particles. Common examples of granularmaterials are sand, gravel, food grains, seeds, sugar, coal, and cement [11,12].

Understanding the behaviour of granular materials is of importance for industry(e.g., storage in silos, transport and chemical processing), energy (e.g., drillingthrough sand for oil recovery) and geophysical processes (e.g., earthquakes anddune formation) and the dynamics of granular media is currently a very activefield of research in physics, mechanics, geophysics, chemistry, pharmacy and civilengineering. Furthermore the processing of granular media consumes roughly 10%of all energy produced on the Earth [13].

Granular materials are conglomerates of particles that interact via contact forcesand can form jammed, solid-like states that resist shear [14]. The flow of granularmaterials has numerous problems and effects in industrial setting. For granularmaterials to flow, shear forces must be applied to overcome repulsive and frictionalcontact forces which hold the particles in static configuration. In steady state, thisimposed energy must be continuously injected into the system and is dissipatedvia inelastic interparticle interactions. The rheology of granular materials is socomplicated due to the effects of inhomogeneity in force transmission. In contrastto continuous media such as Newtonian fluids, forces applied to granular media arealmost resisted inhomogeneities and anisotropic [15].

1.4.1 Dry and wet granular materials

Any child playing on the beach knows that the physical properties of wet anddry sand are very different [16]. The properties of granular materials depends onthe nature of the interaction between the particles and also between the particles

1.4. Granular materials 7

and their environment. When the influence of the environment is negligible andthe interactions are reduced to friction and collisions between the particles, thematerial is in dry granular state. Otherwise, if the solid particles are placed in agas and liquid environment, their behaviour is quite complex and the media areinvolved in the resulting system and we deal with wet granular materials.

Most studies on granular media, have focused on dry granular materials in whichthe dominant interactions are short range and non-cohesive inelastic collisions andfriction. However, many important real life applications involve the mechanicalproperties of wet granular media, such as landslides (which can occur when theground becomes saturated with rain), pharmaceuticals, food processing, miningand waste management and also extremely important properties in geology, agri-culture and civil engineering [17].

Adding a small amount of liquid to granular matter transforms its properties inpractical ways. Wet sand can be easily sculpted [16], and particle agglomeration canbe manipulated, making it an important phenomenon in industrial processes [18].Thus, it is important to study the mechanical response of granular matter withvarious degrees of wetness or liquid content.

The most important effect of a liquid in granular media is the cohesion betweenthe grains. The cohesion occurs in wet granular material before the system turnsto slurry with the liquid or the granular medium becomes completely immersed inthe liquid. In addition to cohesion, there are some velocity dependent phenomena,such as lubrication of the solid-solid friction, which induced by the presence of theliquid and the dissipation behaviour due to the viscosity of the interstitial liquidmay induce velocity dependent behaviour. But in the static regime, cohesion asthe significant difference from dry granular material, plays the most importantrole [17].

One of the effects of cohesion in wet granular materials is that these materials havelow strength against loading. A sandpile is a simple example which shows someeffects of cohesion in wet granular media. Sandpile is a heap formed by pouring afree flowing granular material onto a flat surface. The inclination of the free surfaceof the heap to the horizontal is limited by a maximum value called the angle ofrepose. A granular sample has two characteristic angles: the angle of repose, andthe maximum angle of stability, which is the largest slope a pile can achieve beforefailure occurs. In a dry sandpile, the surface of the pile is smooth and makes a welldefined angle which is less than 40. Also this form of material bears little strengthagainst loading.

All the properties of dry sand are changed by adding a liquid, and just a bit ofliquid enables one to turn a pile of dry sand into a spectacular sandcastle. Theattractive force between the grains, creates a sandpile with a surface angle largerthan that of dry sand and enhances the strength against loading. This angle canbe as large as 90 or even larger which allows for the elaborate construction ofsandcastles (Fig. 1.4 (a)). However too much liquid results in slurry, a form thatcannot keep a shape. Figure 1.4 (b) shows a dry sandpile and a wet sandpile whichcan sustain its structure with a tunnel through it.


Figure 1.4: (a) A sculpture of sand from www.sandpile.de (b) Dry sandpile with a welldefined repose angle and a wet sandpile [17].

1.4.2 Wet granular materials with different degrees of wet-ting

When liquid is added to granular matter, three phases are formed in the system,consisting solid, gas and liquid or interstitial fluid. Interfaces exist between theliquid and solid, and the liquid and gas phases. In dry granular matters, cohesionbetween the grains is negligible while the cohesive strength sharply increases withvery small volumes of liquid because of the roughness of the grains [18]. Thiscohesion arises from surface tension acting on the liquid and gas interface and theLaplace hydrostatic pressure of the bridge which is proportional to the curvature ofthe bridge surface. There are four different regimes of wetting in granular matter[18–20]:

(1) Pendular state: In this regime, there are three phases of liquid, solid and gas (orinterstitial liquid) and particles connect together by liquid bridges at their contactpoints and in the thin liquid film that covers the grain’s surface. The surfaceenergy of those bridges leads to an attractive force between the grains, which isabsent in dry granular materials. In this state the liquid pressure becomes lowerthan the gas pressure and the suction, proportional to the this different pressure(∆P = Pgas − Pliqiud) is positive.

(2) Funicular state: By increasing the liquid content some pores are saturatedby liquid, but there still remain voids filled with gas and ∆P determined by thecurvature of the liquid surface, is small but is still non-zero. In this state the simplebridges between particles merge to give rise to a hierarchy of polyhedral structures.

(3) Capillary state: Almost all the voids between the particles are filled with liquid,but the surface liquid is drawn back into the pores under capillary action and theliquid pressure is lower than the air pressure.

(4) Slurry state: Particles are fully immersed in liquid and no cohesive interactionappears between the particles. the liquid pressure is the same as or larger than theair pressure. These regimes schematically have been shown in Fig. 1.5.

1.5. Suspensions 9

Figure 1.5: Schematic diagrams of the differing regimes of wetting in granular media. (a)Dry grains without cohesion, (b) Pendular regime or partially saturated at small volumefractions such that liquid bridges induce cohesion between the grains, (c) Funicular regimeat higher volume fraction with high degrees of cohesion and liquid bridges merging to givetrimers, tetrahedra and pentamers, (d) Almost saturated or capillary regime such thatlarge wet clusters form (e) The slurry state that the pore space is fully saturated withliquid and no cohesive interaction appears between the particles again [18].

1.5 Suspensions

Suspensions consist of discrete particles distributed in a fluid medium. Generallysuspensions divide into three categories: solid particles in a iquid medium (whichis called suspension), liquid droplets in a liquid medium (emulsion) and gas in aliquid (foam). All these categories have great practical importance, from biologicalmaterials like blood to other industrial dispersions, such as paint and ceramics.Also concentrated suspensions and multiphase polymeric systems are encounteredin many industrial applications. Foodstuffs, pulp and paper and mineral slurriesshow a wide range of these applications [10].

In our experiments, suspension contains solid particles and a fluid in which theparticles lose contact with each other, and are surrounded by fluid. They are in


the category of wet granular materials or in a slurry regime.

Suspensions can illustrate all rheological phenomena from shear-thinning and shear-thickening to time-dependent normal stresses and strong extensional effects. Par-ticle shapes, interparticle forces, and the resulting microstructures are responsiblefor this behaviour. Various forces affect the rheological behaviour of wet concen-trated suspensions. The hydrodynamic forces result from the relative motion ofthe suspending medium with respect to the particles. These are the only forcesconsidered in the Einstein analysis of the viscosity of a dilute suspension of rigidspheres in a Newtonian fluid. Einstein showed that single particles increased theviscosity of a liquid as a simple function of their phase volume:

η = η0(1 + 2.5φ), (1.7)

where η is the viscosity of the suspension, η0 is the viscosity of the suspendingmedium and φ is the volume fraction of suspension. The hydrodynamic forcesare responsible for the migration of the particles. The non hydrodynamic internalforces consist of the Brownian forces, responsible for the internal motion of theparticles and diffusion, and forces arising from the physical and chemical interac-tions. Electrostatic repulsion can occur when particles are charged. Eq. 1.7 is onlyvalid when the interaction between the particles can be ignored, which is valid withvolume fractions of particles up to 10%.

External forces may affect the behaviour of suspensions. Buoyancy effects becomeimportant when the particle and the fluid densities are significantly different. Theseeffects are responsible for sedimentation or flotation, affecting the stability of thesuspensions and resulting in concentration profiles [21,22].

When the density of particles and the fluid are not matched, buoyancy effectsappear and particles denser than the suspending medium will settle under gravity.This is important in viscosity measurements as settling will induce gradients inparticle concentration. Sedimentation is also frequently used in industry as a solid-liquid separation technique. A particle in a dilute suspension, will settle with avelocity V which can be given by Stokes’ law:

V =2R2∆ρg

9η0, (1.8)

where R is the particle size, ∆ρ is the difference between the density of the particlesand the fluid, g is the gravitational constant and η0 is the viscosity of the fluid.It should be pointed out that particles can also be lighter than the suspendingmedium, which produces buoyancy effects. Eliminating this effect in preparing thesamples was one of the important steps in our experiments [4, 17,23].

1.6 Capillary forces between two spherical bodies

Wet granular matter is usually controlled by the interplay among these forces:attraction van der Waals force, gravity, capillary force, electrostatic force or steric

1.7. Basic forces 11

interactions among the particles hydrodynamic interactions and Brownian forces[24]. Capillary forces play a dominant role in wet granular materials and the liquidcreates a network of grains connected by pendular bridges. In section 1.4.2 wediscussed the different regimes of wetting and the effect of the volume fractionof the liquid bridge on the physical properties of wet granular materials. In thissection we investigate the capillary binding between two beads by a pendular liquidbridge.

The capillary forces originate from the axial component of the surface tension actingat the liquid-gas interface and the Laplace hydrostatic pressure in the interior ofa bridge: the Laplace pressure component is proportional to the curvature of thebridge surface. Many works have reported calculations of capillary forces and showthat this force depends on the separation distance of the solid bodies, the surfacetension, geometry, volume of the liquid bridge and the contact angles that theliquid makes with the solid surfaces [25].

Consider Fig. 1.6 showing two beads with the same radii and liquid bridge withsurface tension γ between the gas and liquid. In this Fig. β is the filling angle, andθc is the contact angle between the interface and the surface of the particle. Thesurface tension of the air-liquid interface exerts a horizontal force on the particles asFγ = 2πr1γ. Also the pressure difference between the liquid and gas, ∆p ≡ p1−p2,with a curvature radii r1 and r2 causes an additional horizontal force which isgiven by the Young-Laplace equation ∆p = γ( 1

r1+ 1

r2). The Laplace pressure

is proportional to the curvature of the bridge surface and can be repulsive orattractive. An attractive contribution to the total capillary force occurs if theLaplace pressure is positive and the meniscus is drawn back into the liquid. Thetotal force Fc exerted by the liquid bridge on the particles is given by: Fc =2πr1γ − πr21∆p [17, 25,26].

For the special case of a zero contact angle (θ = 0), the maximum value of attractiveforce is achieved when the particles are in contact, with β → 0, so the capillaryforce is, Fc = 2πRγ.

For particles with a diameter 100 µm and density of about 2.6 gr.cm−3 (sandparticle) and water with γ = 0.07N/m, Fc ≈ 2.2 × 10−5 N, which is about 1600times larger than the weight of the particle. In Chapter 3 we discuss the capillarybridges in granular materials in detail.

1.7 Basic forces

Various forces can describe the force between a pair of colloidal particles in wetgranular materials. Consider two spherical particles which are separated by a dis-tance d and are at rest in the air and an interstitial liquid. The relevant forces actingon these beads are gravity, buoyancy force and the interparticle forces between thebeads including capillary, electrostatic or Coulomb force (between charged parti-cles) and the van der Waals force.

Here, we describe the forces acting on the beads in wet granular materials whichare absent in the slurry state. For the saturated regimes the hydrodynamics forces


resulting from the motion in the liquid medium are important while in the capillaryor pendular regime, the liquid does not fill all voids in the material and cohesionforce between them is dominant.

The attractive van der Waals force plays an important role in calculating the forceswhich act on the particles. The origin of this force is ultimately quantum mechan-ical, arising from the interaction between fluctuating dipoles in atoms. For macro-scopic bodies, the interactions among the molecules of two bodies must be suitablysummed to obtain the total van der Waals force. For two identical spherical bodieswith radius R and separated by a distance d, the van der Waals force is given by:

Fv = − AR

12πd2, (1.9)

where A is known as the Hamaker constant. The Hamaker constant is a materialproperty and for a lot of materials its order of magnitudes is around 10−19 J. Thisequation is not valid for large values of d (d > 5 nm), as the force decays morerapidly than is assumed in the derivation of Eq. 1.9. So we do not observe thisforce for macroscopic bodies like sands, because their surfaces are too rough toobtain enough intimate contact. Also, the Coulomb force exerted by a sphericalcharged body on the same body with the separating distance d is given by:

Fcoulomb =q1q2

4πε0ε(d+ 2R)2, (1.10)

where q1 and q2 are the total electric charges on the spherical bodies, ε0 is thepermittivity of free space, ε is the relative permittivity or dielectric constant, and(d + 2R) is the distance between the centers of the spheres. In (SI) units, ε0 =8.85× 10−12 C2/Nm2.

When the air contains moisture, an adsorbed layer of liquid molecules forms onthe surfaces of the particles at a low humidity. At high humidity, condensationof the water leads to the formation of a liquid bridge and the attractive capillaryforce between the grains. The force associated with a liquid bridge is discussed inChapter 3 in more detail. But for a simple estimation of this force, we considera layer of liquid which connects two particles as a bridge (Fig. 1.6). The surfacetension of the air-liquid interface exerts an attractive force between the particles.In addition, the pressure difference between liquid and air (or the curvature of theinterface) causes an additional force to be exerted on the particles. The consequenceof these forces is the capillary force which for two spherical bodies with radius Rin the limit of particles in contact with separated distance d = 0, can be simplifiedas:

Fc ∼ 2πRγ. (1.11)

Now we can estimate the order of magnitude of these forces for two sphericalparticles of radius R = 50 µm, particle density ρ = 2600 kg.m−3, separationd = 1 nm, charge density 10−5 Cm−2 and dielectric constant of air ε = 1. Theweight is 1.4 × 10−8 N, the Coulomb force is about 8.9 × 10−8 N and the van

1.8. Brownian motion 13

Figure 1.6: Schematic of a liquid bridge (blue) between spherical surfaces. θ is thecontact angle between the air−liquid interface and the surface of the particle, and β isthe filling angle.

der Waals force is 1.3 × 10−7 N. Meanwhile, the capillary force, considering thesurface tension γ = 0.07 N.m−1, is about 2.2× 10−5 N which is much larger thanthe van der Waals, weight and electrostatic forces. Also, we discussed that thevan der Waals force for the rough surfaces goes to zero. The electrostatic forceplays an important role in dry experiments. In wet systems with an electricallyconductive liquid we can ignore the electrostatic force. So by considering the size ofthe particles, we can find the dominant forces between the particles in wet granularmaterials. In the experiment in this thesis, the important and dominant force isthe capillary force [23,27–29].

1.8 Brownian motion

Brownian force acting on a colloidal particle arises from the random thermal colli-sions of the suspending medium with the colloidal particles and leads to diffusivemotion. For particles of the order of less than 10 µm in size, Brownian motionintroduces an effective force that acts to keep particles well distributed. A dimen-sionless number known as the Peclet number (Pe), relates the shear rate of theflow (γ) to the particles diffusion rate. The Pe number can be defined in terms ofthe applied shear stress σ:

Pe =γR2

D=σR3

kT, (1.12)

where R is the size of the particles and the diffusion coefficient, D, is definedaccording to the Stokes−Einstein−Sutherland equation, D = kT/6πη0R. Eq. 1.12


defines the limitation of the Brownian regime in the colloidal suspensions. The Penumber, useful in dispersion rheology, is often measured by applied shear rates orshear stresses. In a low Pe number, the system is close to equilibrium in which theBrownian motion can largely restore the equilibrium microstructure on the timescale as the slow shear flow. At high shear rates or stresses, deformation of thecolloidal microstructure by the flow occurs faster than that which Brownian motioncan restore. At Pe ∼ 1, particles become ordered by the flow and can be organizedinto layers that are able to flow with less resistance, reducing the suspensionsviscosity. This effect, known as shear-thinning, is defined by the behaviour suchthat, at high shear rate, the shear stress is σ ∼ γn with n < 1; so the viscositydecreases as the shear rate increases. As the Pe number increases, hydrodynamicinteractions can destabilize the order of the particles. In this case, the increasednumber of collisions in the disordered state leads to a regime of shear-thickening,where the viscosity dramatically increases and particles organize into hydroclusters(Fig. 1.9 (b)). Also, according to Eq. 1.12, the size of the colloidal particles is oneof the important parameters which defines the transition from Brownian suspensionto non-Brownian colloidal suspensions regimes. In this thesis, all the experimentswere carried out with particles larger than 20 µm, so thermal noise plays no rolein the particle motions and the Brownian motions are negligible.

1.9 Non-Newtonian behaviour

For the Newtonian materials the stress tensor σij is expressible as a linear func-tion of the rate of deformation tensor γij so the viscosity is constant and the twonormal stress differences are zero. Otherwise, the fluid is called a non-Newtonianfluid. Thw material properties can be conceived as being two limiting extremes atelastic (Hookean behaviour) and viscous (Newtonian behaviour). The stress-straincurves show the elastic response of the materials to the applied forces. For a non-Newtonian fluid, the flow curve (shear stress vs. shear rate) is non-linear or doesnot pass through the origin and the viscosity is dependent on the flow conditions,such as flow geometry and shear rate.

For complex fluids, the relation between shear stress and shear rate depends, inaddition, upon the duration of shearing and their kinematic history. So two condi-tions are crucial in realizing the nonlinear behaviour of these materials. The firstis that the strain experienced by the material is large enough so that the theoryof linear elasticity no longer applies. The second is that the Deborah number isnot small. The Deborah number is the ratio of the material relaxation time to thecharacteristic time of the deformation process being observed, D = τ/T . The timeτ , is infinite for an ideal Hookean elastic solid and is zero for a Newtonian viscousliquid. So when the Deborah number is very small, as in rapid deformation, theelastic solid approximation is appropriate for these materials (like polymer melts).However polymer gels and associated colloids have longer relaxation times (morethat 100 seconds). In this section, we investigate deviation from the Newtonian

1.9. Non-Newtonian behaviour 15

Figure 1.7: Demonstration of shear viscosities for a shear-thinning of a polymer solution[30,31].

behaviour in materials, called shear-thickening, shear-thinning and yield-stress ma-terials.

1.9.1 Shear thinning and shear thickening

When experiments were carried out to investigate the effects of the shear rateon viscosity, scientists observed some deviations from Newtonian behaviour formany materials such as polymer solutions, emulsions, gels and suspensions. Shear-thinning is a non-linear phenomenon in which the viscosity decreases with increasein the shear rate. Figure 1.7 shows the behaviour of viscosity versus shear ratein a shear-thinning material. The figure indicates that in the limit of very lowor very high shear rates (or stresses), the viscosity is constant, but at differentlevels. These two extremes are sometimes known as the lower and upper Newtonianregions, respectively. These levels of viscosities depend on factors like the type andconcentration of the polymer, its molecular weight distribution and the natureof the solvent. Also, in the shear-thinning region, the normal stress componentsare not equal and the differences increase with the shear rate which shows thenonlinear behaviour the in shear-thinning regime. Shear-thickening is defined asthe increase in viscosity with shear rate. Most concentrated suspensions exhibitshear-thickening at high shear rates and this type of fluid behaviour is originallyobserved in concentrated suspension. The onset and the importance of shear-thickening depends on the volume fraction, particle size particle distribution andviscosity of the suspending fluid, but the detailed mechanism of shear-thickeningis still under debate.


Figure 1.8: Schematic representation of viscosity vs. shear rate for a shear-thickeningsystems, for different volume fraction of suspensions [32]. The green points show thecritical shear rate γc, at which the viscosity starts to increase and the red ones show theshear rate at which the maximum viscosity is reached; γm. These results show that thecritical shear rate decreases as the phase volume of the suspension increases.

Understanding the shear-thickening phenomena is an important subject in the topicof non-Newtonian behaviour. It is observed in dense colloidal suspensions, whereit has been related to the formation of dense clusters of particles. The shear-thickening phenomenon can damage processing equipment and induce dramaticchanges in suspension microstructures and industrial processes, such as particleaggregation, coating qualities, fouling pipes and spraying equipment. Because ofits negative impact on industrial processes, shear-thickening has been extensivelystudied over the last three decades [33]. Also given the correct conditions, all con-centrated suspensions of non-aggregating solid particles will show shear-thickening.

Figure 1.8 shows the shear-thickening behaviour for a range of particle concentra-tions [4]. These systems have a high but finite viscosity at the zero shear rate. Atlow shear rates, the viscosity begins to decrease; at much higher shear rates theviscosity tends to be smooth; and above a critical shear rate (γc), the viscosity in-creases. After this shear-thickening region the viscosity achieves a maximum valueand can level out to a new plateau value. This Newtonian plateau region, or thespecific situation shown in Fig. 1.8 in which the viscosity has decreased again, iscommented on by Green, Griskey and Hoffman [32].

An explanation for the increase in viscosity is the transition from a two-dimensionallayered arrangement of particles to a random three-dimensional form. The two-dimensional arrangement is brought by the flow rearranging the particles intoclosely packed sheets flowing over each other. This layered arrangement gives alow as possible viscosity for a suspension. The layered arrangement is unstable,and is disrupted above a critical shear stress which causes a random arrangementand increases the viscosity. This effect was studied using an optical diffractionsystem by Hoffman in 1972. Before shear-thickening occurs, particles move in lay-

1.9. Non-Newtonian behaviour 17

Figure 1.9: (a) The fluctuation in microstructure to show the transitions to shear-thinning and shear-thickening. In equilibrium, random collisions among particles makethem naturally resistant to flow. But as the shear stress (or, equivalently, the shear rate)increases, particles become organized in the flow, which lowers their viscosity. (b) Thechange in microstructure of a colloidal dispersion and transitions to shear-thinning andshear-thickening: In equilibrium, random collisions among particles make them naturallyresistant to flow. But as the shear stress (or, equivalently, the shear rate) increases,particles become organized in the flow, which lowers their viscosity. At higher shearrates, the hydrodynamic interactions between the particles dominate over the stochastic,and particles aggregate into closely connected clusters (red). The difficulty of particlesflowing around each other in a strong flow leads to a higher rate of energy dissipation andan abrupt increase in viscosity [34].

ers, but they do not have to be rigorously ordered in these layers, and the layerscan have a thickness ranging from one particle diameter to some multiple of theaverage particle diameter. Then, at the onset of shear-thickening, a hydrodynamicinstability acts to drive particles out of these layers. When this happens, theparticles interact through clustering by physical contact. Lubrication forces andfrictional forces are most likely involved in this shear-thickening process in one caseor another [35].

Figure 1.9 shows the viscosity of colloidal dispersions and the transition betweena shear-thinning and shear-thickening in that regime. This phenomenon was onlyobserved at a volume fraction greater than a critical value φc. At high particlevolume fractions, the fluid shows an apparent yield-stress. When the yield-stressis exceeded, the viscosity of fluid decreases, a response known as shear-thinning.At higher stresses, shear-thickening occurs: viscosity rises abruptly once a criticalshear stress is reached. As explained above, Hoffman developed a micromechanicalmodel of shear-thickening as a flow induced order-disorder transition.

Another origin of shear-thickening can refer to the appearance of normal stresseswhich are usually associated with concentrated polymer solutions. Each shear-thickening suspension exerts a force on the plates rheometers which are used tomeasure the effect. Windle and Beazley, Willey and Macosko, and Kurath and


Larson give examples of the appearance of normal stress differences coincidentwith shear-thickening. The latter, using die swell measurements, found that theindicated normal stress was proportional to the shear rate squared. It seems quiteclear, that often shear-thickening and normal forces go together, and that althoughgenerally smaller than the normal forces found in concentrated polymer solutions,they can be quite significant in high speed industrial processes [32].

1.9.2 Yield stress

Yield-stress materials have been studied for about a century, initiated by Bingham.These materials behave as solids under small stresses but flow like liquids beyonda critical stress. In the classical description, initiated by Bingham, there is noflow (infinite viscosity) if the shear stress is less than a critical value (the yieldstress) and the stress is a monotonically increasing function of the shear rate abovethat value. The canonical yield stress picture assumes that the material is soliduntil a critical shear stress is exceeded and above that critical stress the materialsubsequently flows. So yield-stress materials will not flow until the applied shearstress is exceeded the yield stress (σy) and below this critical value any deformationproduced by an external force will be purely elastic. The simplest model to describefluids exhibiting a yield-stress is the Bingham model which is given by the followingequation [36–38]:

σ = σy + ηpγ, (1.13)

where σy is the yield-stress that shows the transition from elastic to plastic defor-mation and is a limiting shear stress at which the material starts to flow and ηp isthe plastic viscosity. These descriptions show that a yield-stress material effectivelybehave as a solids for applied stresses below a critical yield stress σ < σy but irre-versibly deform and flow as a fluid for applied stress above the yield stress σ > σy.There are another models widely used to describe such behaviour of yield-stressfluids like the Herschel-Bulkley (H-B) model, σ = σy + kγn, where k and n areadjustable model parameters.

The yield-stress is a useful rheological parameter to characterize particle-particleinteractions in suspensions. It is important to understand their flow characteris-tics, and notably the yield-stress, to correctly predict the flow behaviour of thesematerials.

In many systems, such as clay suspensions, the yield-stress is due to an internalstructure that is broken down by the flow. As the microstructure is degraded, theviscosity may drop rapidly, and the stress required to initiate flow may then be muchhigher than that required to maintain it [38, 39]. Also wet granular material is anexample of yield-stress fluids in which, above a critical yield-stress, their mechanicalresponse changes from a solid-like to a dissipative flow. complex fluids such asdense suspensions or polymer gels that exhibit a critical yield-stress have extremelyvariable viscoelastic properties ranging from elasto-plastic behaviour below theyield-stress to viscous behaviour beyond that. Thus, the linear viscoelastic domain

1.10. Rheometry; experimental techniques 19

of yield-stress materials is typically limited to very low strain, and is not relevantto many industrial applications [40].

The techniques developed for measuring yield-stress can be categorized into twogroups which involve the fitting of the shear stress/shear rate data to rheologicalmodels. The yield-stress is determined by the extrapolation of the flow curve at lowshear rates to zero shear rate [40]. The results obtained using other methods arevery sensitive to the models used for fitting the rheological data and the accuracyof the rheological measurement in the low shear rate region [41].

1.10 Rheometry; experimental techniques

A rheometer measures the rheological properties of a complex fluid as a function ofrate or frequency of deformation. Due to the related applications of the rheometers,they divide into two types: drag flow in which the imposed shear on the fluidis generated between moving and fixed surfaces and pressure flow in which theimposed shear is generated by a pressure difference over a closed channel. All thesegeometries can be used to measure shear material parameters: elastic modulus G,viscoelastic functions, like loss and storage modulus G∗, viscosity η, normal stressdifferences N1 and N2 and other parameters.

In this chapter we explain the first group of these rheometers; the cone-plate,parallel plate and concentric cylinder rheometers and in this thesis most of theexperiments have been undertaken using plate-plate geometry.

1.10.1 Concentric cylinder rheometer

The first rotational rheometer was made by Maurice Couette in 1890. This kindof rheometer consists of two concentric cylinders with a gap between them [10].Figure 1.10 shows a schematic of this rheometer with the inner radius r1 and outerradius r2.

If we consider a steady and laminar flow, confined between two concentric cylinderin which the inner one rotates at angular velocity Ω, the velocity field of the fluidin the cylindrical coordinate is given by:

~υr = ~υz = 0; ~υ = ~υθ = rΩ. (1.14)

Then the shear stress and shear rate distribution across the gap between the cylin-ders is obtained by solving the Navier-Stokes equation. For a small gap the shearrate and shear stress parameters are given by:

γ =r2Ω

r2 − r1, (1.15)

σ =T

2πr22L, (1.16)

where T shows the applied torque on the cylinder and L is the immersed length ofthe liquid being sheared [10].


Figure 1.10: Schematic of a concentric cylinder rheometer.

1.10.2 Cone and plate geometry

In 1934 Mooney and Ewart suggested cone-and-plate geometry for viscosity mea-surements. In cone-and-plate geometry, the fluid is sheared between a plate and acone with a small angle θ. The small cone angle (θ < 4o) ensures that the shear rateis nearly uniform throughout the shearing gap (Fig. 1.11). Today this geometry,with its constant shear rate and direct measurement of normal stress difference, N1,is probably the most popular rotational geometry for studying the non-Newtonianeffects of materials.

Assuming the plate is stationary and the cone is rotating with an angular velocityof Ω. The spherical coordinate is an appropriate choice to describe the velocityfield of the fluid at any point in this geometry:

~υr = ~υφ = 0; ~υ = ~υθ(r, φ). (1.17)

Also the shear stress and the shear rate can be found by solving the Navier-Stokesequation:

γ =Ω

tan θ(1.18)

σ =3T

2πR3. (1.19)

where R is the radius of the cone and T is the torque of the cone.

In Newtonian materials the only stress generated in simple shear flow is the shearstress, so the two normal stress differences are zero: N1 = N2 = 0. But in the

1.10. Rheometry; experimental techniques 21

Figure 1.11: Cone-and-plate geometry.

viscoelastic materials the microstructure of the fluid is anisotropic, the distribu-tion of shear stress is more complicated, and we need to define the normal stressdifferences. This anisotropy leads to a normal force F which acts on the cone inthe direction of the axis of rotation and pushes the cone and plate apart. So, thefirst normal force difference is given by:

N1 =2FzπR2

. (1.20)

Considering the inertia effect in calculating the normal stress difference leads us tothe “negative normal stress effect”, whereby the plates are pulled together and themeasured value of the force F is smaller than the correct value. This reduction inforce F is given by:

∆Finertia = 0.075πρΩ2R4, (1.21)

where ρ is the density of the fluid. By combination of Eq. 1.20 and 1.21, we obtaina correct measure of N1:

N1 =2FzπR2

− 0.15ρΩ2R2. (1.22)

So, first the normal stress difference can be obtained from the axial force by cone-plate geometry and second the normal stress difference can be measured by plate-plate geometry.

The main advantages of a cone-plate rheometer are the homogeneous shear field(for cone angles up to θ < 4o), which leads to easier data analysis than that ofthe coaxial cylinders and parallel plate devices, and also the small sample volumesrequired compared to the cylinder rheometer. But, this geometry is not suitable for


Figure 1.12: The parallel plate geometry.

suspensions due to the possibility of particle jamming. In summary, cone-and-plategeometry is suitable for materials which need to be subjected to a uniform shearrate (especially for low to medium viscosity materials) such as sensitive foodstuffs,care products and so on. This geometry is also frequently used for measuring thefirst normal stress difference [10].

1.10.3 Parallel plate geometry

The paralle plate geometry was suggested by Mooney in 1934. The Mooney testerwas consist of a disk rotating inside a cylindrical cavity [10]. In 1946, Russellundertook the first experiments on this geometry and also on cone-plate geometryto measure the normal forces. A typical example of such an instrument is shownin Fig 1.12. It comprises two parallel plates separated by a distance h.

We assume a laminar, isothermal and steady flow, between two parallel disc, inwhich one of them is stationary and the other rotates with the angular velocity Ω.The velocity field of the fluid in the cylindrical coordinate is given by:

~υr = ~υz = 0; ~υ = ~υθ(r, z). (1.23)

Solving the Navier-Stokes equation with a no-slip boundary condition at the sur-faces and ignoring inertial forces, the shear stress and shear rate are given by:

γ =RΩ

h, (1.24)

σ =3T

2πR3, (1.25)

where R is the radius of the plates, h is the gap between the plates and T is thetorque of the rotation of the plate. It can also be shown that

1.11. Outline of the thesis 23

(N1 −N2) ˙γR =F

πR2(2 +

d lnF

d ln γR), (1.26)

F is the total normal force which arises from the anisotropic structure of the fluidexerted on the plates. The inertia effects are similar to those of the cone-and-plate.Total force data yields the combination N1 −N2 in the plate-plate and also N1 inthe cone-plate geometries.

One advantage of this geometry is that the shear rate can be changed by bothΩ and the gap, and high shear rates can be attained with small gaps. In theseexperiments, an external normal force is applied to the upper plate and the gaph is allowed to vary until this force balances the normal force generated by thesample between the plates. Gap h is measured and Eq. 1.26 applied [10].

1.11 Outline of the thesis

This thesis comprises two parts and five chapters. The first part concerns the‘Rheology of dry and partially saturated granular materials’ and the second isdevoted to the ‘Rheology of wet granular materials’. The first part is organized asfollows:

Chapter 2: In this chapter we investigate the non-linear rheological responseof sand with and without small amounts of liquid under steady and oscillatoryshear. By using two very different set-ups or techniques (quasi-statically push thesand and large amplitude oscillatory shear), we study the flow of dry and partiallysaturated sand. Our results show that the resistance against deformation of thepartially saturated sand is much smaller than that of the dry sand, and the drysand dissipates more energy under flow.

Chapter 3: Here we review the effects of the liquid content in the granular ma-terials and capillary bridges between the grains. We study the mechanism of thebuckling and the buckling of a column under its own weight. Furthermore, the sta-bility of wet sand columns has been accounted for by measuring the elastic modulusof the wet sand. The results suggests that the maximum height of the sand columnis related to the base radius of the sand column as the 2/3 power.

The second part of the thesis is devoted to the rheology of completely wet granularmaterials and we aim here to study the behaviour of suspensions which are in theslurry state or fully saturated in the regime of wetness.

Chapter 4: This chapter focuses on the study of irreversibility of the viscous flow.We investigate the homogeneity of the suspensions by Magnetic Resonance Imagingand show an irreversible migration of particles from high-into low shear rate regions.Also above a critical deformation, large normal forces appear in the suspensionswhich arises from frictional contacts between the particles. Such contact also leadsto irreversibility in the motion of the particles.


Chapter 5: This chapter aims to study the behaviour of dense suspensions of non-Brownian particles. Measurements were performed with an rotational rheometer inconfined geometry (cup-plate) under imposed gap. We investigate the emergenceof shear-thickening in the density-matched suspensions with diameter 20 µm in thewater. We find that the local rheology presents a transition at low shear rate froma viscous to a shear-thickening behaviour with shear stresses proportional to theshear rate squared, as predicted by a scaling analysis.

Part I

Rheology of Dry and PartiallySaturated GranularMaterials

25

2.

Dissipation in wet and drygranular materials

Abstract: In this chapter we review viscoelastic properties of materials and ex-plain different experimental methods of studying the linear and non-linear be-haviour of matter under shear tests. Then we investigate the stress-strain behaviourof sand with and without small amounts of liquid under steady and oscillatoryshear. Since dry sand has a lower shear modulus, one would expect it to deformmore easily. Using a new technique to quasi-statically push the sand through atube with an enforced parabolic (Poiseuille-like) profile, we minimize the effect ofavalanches and shear localization. We observe that the resistance against defor-mation of the wet (partially saturated) sand is much smaller than that of the drysand, and that the latter dissipates more energy under flow. This is also observedin large amplitude oscillatory shear measurements using a rotational rheometer,showing that the effect is robust and holds for different types of flow.

2.1 Linear viscoelastic behaviour

Viscoelasticity is a result of simultaneous viscous and elastic properties in a ma-terial. Understanding the rheological behaviour of complex suspensions is an es-sential step in understanding the technological significance of their viscoelasticproperties [4, 42].

Development of the mathematical theory of linear viscoelasticity is based on asuperposition principle. This implies that the response (e.g. strain) at any time isdirectly proportional to the value of the initiating signal (e.g. stress). In the lineartheory of viscoelasticity, the differential equations are linear. Also, the coefficientsof the time differentials are constant. These constants are material parameters,such as the viscosity coefficient and rigidity modulus, and they are not allowed tochange with changes in variables such as strain or strain rate. Further, the time

27

28 Chapter 2. Dissipation in wet and dry granular materials

derivatives are ordinary partial derivatives. This restriction has the consequencethat linear theory is applicable only to small changes in the variables. We can nowgenerate a general differential equation for linear viscoelasticity as follows:

(1 + α1∂

∂t+ α2 ∂

2

∂t2+ ...+ αn

∂n

∂tn)σ = (β0 + β1

∂

∂t+ β2

∂2

∂t2+ ...+ βm

∂m

∂tm)γ (2.1)

Some important special cases of Eq. 2.1 follow. For example if β0 is the onlynon-zero parameter, σ = β0γ, which is the equation of Hookean elasticity and β0is called the rigidity modulus, G. For the only non-zero parameter β1, we have theNewtonian viscous flow and this parameter shows the coefficient of viscosity, η.

If just both α1 and β1 are non-zero, the Eq. 2.1 leads to the “Kelvin model” whichis one of the simplest models of viscoelasticity:

σ = Gγ + ηγ. (2.2)

Another simple model is the “Maxwell model”. The differential equation for themodel is: σ + α1σ = ηγ, where α1 = τM = η/G called the relaxation time.

The next level of complexity in the linear viscoelastic behaviour is to make otherparameters of Eq. 2.1 non-zero. These kind of equations can be derived mathemat-ically for different suspensions with viscoelastic behaviour. In addition, many otherideas have been employed to develop elementary models for viscoelastic behaviour.The response of the material to a linear or constant input parameter (e.g. stressor strain) is obtained by solving Eq. 2.1 and its variants.

2.1.1 Oscillatory shear tests

A key to measuring the linear and non-linear viscoelastic behaviour of the materialsis the determination of their response to small amplitude oscillatory shears (SAOS),which weakly perturb the equilibrium structure, and large amplitude oscillatoryshears (LAOS) [43,44].

Dynamic oscillatory shear tests are common in rheology and have been used to in-vestigate a wide range of soft matter and complex fluids including polymer melts,biological macromolecules, polymers, surfactants, suspensions, emulsions and be-yond. Although small amplitude oscillatory shear is a popular deformation modein probing the linear viscoelastic behaviour of materials, the useful properties ofsoft materials are often related to their responses to high strains [1, 45,46].

Large amplitude oscillatory shear (LAOS) is a class of flow that is commonly usedto characterize nonlinear viscoelastic material behaviour. The first studies of LAOSrefer to early publications in the 1960s to 1970s. The methods for analysing LAOSinclude Lissajous curves (Philippoff 1966; Tee and Dealy 1975), Fourier transformrheology (e.g., Wilhelm 2002), stress decomposition (Cho et al. 2005; Ewoldt etal. 2008; Yu et al. 2009), and computation of viscoelastic moduli (Hyun et al.2002; Ewoldt et al. 2008). In this thesis, the physical meaning of the LAOS

2.1. Linear viscoelastic behaviour 29

measurements is highlighted by considering graphically the raw test data in theform of Lissajous curves, which are plots of stress σ(t) vs. strain γ(t) [44].

Both the elastic and viscous characteristics of a material can be examined simul-taneously by imposing an oscillatory shear strain. An oscillatory strain at fixedfrequency is applied to the sample and consequently subjects the sample to a cor-responding oscillatory strain rate:

γ(t) = γ0 sin(ωt), γ(t) = γ0ω cos(ωt), (2.3)

where ω is the frequency of rotation and γ0 is the strain amplitude which is smallenough for the linearity constraint to be satisfied. The strain amplitudes used inlinear oscillatory shear tests are very small, often of the order of γ0 ≈ 10−2 and forsome emulsions or suspensions the linear regime is limited to even smaller than thisvalue. Then the stress response to this input deformation σ(t;ω, γ0) is recordedand analysed. So the LAOS measurements are defined by two input parameters,frequency and amplitude ω, γ0 and the response of the material in terms of stresscan be written as:

σ(t) = γ0[G′ sin(ωt) +G′′ cos(ωt)]. (2.4)

When γ0 is constant, the linear viscoelastic response is observed. For ideal solidsthat follow Hooke’s law, the shear strain is proportional to the shear stress and thefunctions are in phase (Fig. 2.1 (a)). For the ideal fluids that follow Newton’s lawof viscosity, the shear rate is proportional to the shear stress with the phases shiftedby 90o (Fig. 2.1 (b)). For the viscoelastic materials the shear stress function canbe separated into an elastic stress in phase with the shear strain, and a viscousstress in phase with the shear rate (Fig. 2.1 (c)).

The storage modulus is the ratio of the elastic stress to shear strain while the lossmodulus is the ratio of the viscous stress to shear strain. At low strain amplitudeswhen the response is linear, the material is commonly characterized by the vis-coelastic moduli G′ and G′′ as determined from the components of the stress inphase with γ and γ, respectively. The constant storage modulus G′ and the lossmodulus G′′ are defined in the linear viscoelastic regime, and their values at highstrain amplitude need care in the measurements to find the G′(γ0) and G′′(γ0).

The dependence of storage and loss moduli on strain amplitude is interpreted asa non-linear viscoelastic response. The material’s resistance to deformation is thecomplex shear modulus G∗ consisting of an elastic or storage modulus (G′) and aviscous or loss modulus (G′′):

G∗ = G′ + iG′′. (2.5)

The oscillatory test responses can be described as Lissajous curves with stressversus strain that provide a meaningful way to visualize and understand viscoelasticbehaviour in general. A linear elastic material response, σ = Gγ, appears as astraight line on the Lissajous curve of σ(t) versus γ(t) and in the linear viscoelasticregime, the Lissajous figures are elliptical when the stress response is a sinusoidalfunction with a phase difference 0 < δ < π/2 compared with the strain (Fig. 2.1).


Figure 2.1: Plot of the imposed sinusoidal stress function (dashed) and recorded shearstrain (solid) in dynamic oscillatory tests for an (a) ideal elastic, (b) viscous and (c)viscoelastic material. σ(t) is the dynamic shear stress function, σ0 is the amplitude ofthe stress function, γ(t) is the dynamic shear strain function, γ0 is the amplitude of thestrain function, ω is the frequency of the imposed oscillation, and δ is the phase shiftangle between shear stress and shear strain [47].

The strain amplitude γ0 can be increased systematically to enter the non-linearviscoelastic regime. A non-linear viscoelastic response will change the ellipticalshape of a Lissajous curve with the deviation from linear viscoelastic behaviour.When the strain amplitude γ0 becomes large, for an oscillatory strain input, γ(t) =γ0 sin(ωt), the viscoelastic stress response can be written as a Fourier series of oddharmonics:

σ(t;ω, γ0) = γ0∑n:odd

G′n(ω, δ0) sin(nωt) +G′′n(ω, δ0) cos(nωt). (2.6)

For sufficiently small strain amplitude γ0, a linear material response is observedsuch that only the fundamental harmonic appears, n = 1 with a temporal phaseshift δ1 given by tan δ1 = G′′1/G

′1 [48,49]. For larger deformation amplitudes, higher

2.2. Flow of wet and dry granular materials 31

Figure 2.2: Oscillatory strain sweeps of pedal mucus gel from Limax maximus [50] ata frequency ω = 3 rad.s−1 which shows the plotting of the raw data of σ(t) versus γ(t)reveals nonlinear characteristics of this gel. Inset: the linear behaviour in low strain [48].

harmonics appear, and the response is nonlinear.

Figure 2.2 shows the raw data and the corresponding non-linear behaviour of σ(t)and γ(t). In this work, a series of oscillatory tests at a fixed frequency of ω =3 rad.s−1 are imposed and the corresponding nonlinear behavior of the physicallycross-linked mucus gel is shown [47]. The inset shows a linear viscoelastic responseappearing as an ellipse for small γ0, which contains the major and minor axes ofthe ellipse and the nonlinear viscoelastic response loses the elliptical shape [44,45,48,49].

2.2 Flow of wet and dry granular materials

In Section 1.4.2 we discussed the different regimes of wetting in granular matterand showed that the small amount of liquid added to a dry granular material whichforms “bridges” at the contact points between the grains, changes the mechanicalproperties of these matters. In Chapter 3 we investigate dry granular media andalso the impact of the amount of fluid in the granular media and its effect onthe strength of a wet granular material. For dry sand, a major step in describingthe rheological properties was the introduction of the Coulomb friction approach


[51–53]. This relates the shear stress to the confinement pressure via a frictioncoefficient [54].

Wet (partially saturated) granular materials have been studied mostly in the geo-physics literature since soils are a typical example of such a system. Here, thecharacterization of flow properties is very different; these materials are reported toexhibit a mixed behaviour of elasticity, viscosity, and plasticity. The system startsto flow when the externally imposed stress exceeds the inter-aggregate contactforces [55]. The mechanical properties at lower water content are determined bythe liquid bridges between grains, and those at higher water content are determinedby the flow of the liquid through the soil pores [56].

In this chapter we compare the properties in incipient flows of wet and dry granularmaterials near the jamming transition. Partially saturated granular matter has amuch higher yield-stress (allowing the construction of a sand castle) and shouldtherefore have a much higher apparent viscosity for slow flows [57, 58]. It meansthat above a critical yield-stress, the mechanical response changes from a solid-liketo a dissipative flow. The microscopic mechanism for this behaviour refering to thereorganization of individual particles that resists the motion and the microstructurepresent in the fluid that resists large rearrangements is responsible for the yield-stress. For this reason, it is commonly believed that wet sand should show a largerresistance to flow, i.e., more viscous, than dry sand [18,26,59].

We will show, however, that in two very different set-ups, the energy dissipation, islarger than that of wet sand. We show that this is due to the fact that the adhesionbetween the grains decreases the confining pressure and hence decreases the flowresistance.

2.3 Experimental techniques

2.3.1 Shear cell

This experiment were carried out using sand composed of glass spheres of diameterd = 140−150 µm with and without additional deionized water. The water contentω is defined as the ratio between the liquid volume and the volume occupied bythe grains. The experiments were carried out with the shear cell described inreferences [19, 60]. The experimental works relevant to the shear cell have beencarried out in collaboration with the Saarland University. The use of this devicemakes it possible to shear the granular materials homogeneously over the samplevolume.

The granular material is put into an acrylic cylindrical cell (Fig. 2.3(a)). Acylindrical cell made of acrylic glass and with a diameter equal to its lengthL = D = 24 mm, was filled with the sample (brown). The “flat sides” of thecell each consist of a thin latex membrane with a thickness of 300 µm. Adjacent toeach membrane was a cylindrical chamber filled with water connected to a syringe.So the sample is separated from the confining water (in the chamber) by the flexible

2.3. Experimental techniques 33

Figure 2.3: (a) Tube experiments measurement cell containing granular matter (brown);a volume change ∆V provokes a difference between p1 and p2, the pressures in the ad-jacent chambers; D and L, are the cell diameter and length, respectively. (b) Cup-platerotational rheometric set-up.

membrane. Also the membrane transfers the presure to the confined sand in thecell.

Dry sand gently was poured into the cell and the density corresponding to that ofrandom dense packing φ = 0.63, was achieved by tapping the sand in the cell. Thecell was also filled with wet sand with 0.01 ≤ ω ≤ 0.3 and φ = 0.63. It was possibleto inject water into and extract it from the chamber through tubes via a syringe.The syringe pistons were connected to spindles that could be moved with a stepmotor.

When the two pistons were moved at equal speed and in opposite directions, thesample between the membranes was deformed at constant volume and well definedspeed. The tension of the membranes had approximately parabolic deformation,and and imposes a Poiseuille-like profile within the sample. In this way, avalanch-ing was kept to a minimum, and a rather homogeneous shear deformation of thesample is achieved. Piezoresistive sensors measure the pressure (with respect to at-mospheric pressure) in both chambers. A fixed quantity of water was then injectedinto or extracted from the adjacent chambers, and the differential pressure p1− p2was measured. A family of differential pressure characteristics for increasing shearvolumes is shown in Fig. 2.4.

It is worth mentioning that for a given shear amplitude, the hysteresis loop isstable during as many cycles. A few tests were carried out which were one weekin duration and found that the opening of the loop for different shear amplitudesremains constant (we made sure that all our data were taken in such a steady-state situation). Also the shear rates γ were tested between 0.0002 and 0.1 s−1

without observing differences in the hysteresis loop. The measured quantities for


Figure 2.4: Differential pressure curves with increasing shear amplitudes for wet sand.The measured parameters for each curve are σ and ∆V .

each differential pressure curve are the opening of the loop at zero shear σ and thevolume ∆V . The imposed profile permited us to estimate the equivalent for thewall shear strain for the cylindrical cell as γmax = 32∆V/πD3 [61].

Strain γ is the input parameter which we apply on the system and then we ob-tain the response of the material to this excitation, which is shear stress. In theoscillatory form of deformation (or strain-controlled deformation) we have an re-ciprocating motion which in these tests the strain changes by time according to thestrain amplitude during a cycle. So we can define the maximum value of γ duringa loop of a reciprocating motion.

2.3.2 Oscillatory shear tests

For a better understanding of the results and to check their robustness, I performeda similar experiment in a completely different geometry. This approach emphasizesthe visual representation of the LAOS stress response and enables us to explorehow the material properties characterizing the dessipated energy depend on size ofthe grains and liquid volume fraction. Also oscillatory measurements are suitabletests to characterize the macroscopic response related to the microstructure of amaterial.

For this, I used a standard rheometer with a geometry that allows a test underconfinement. I used cup-plate geometry (Fig. 2.3 (b)) and studied polystyrenespheres (Dynoseeds) with diameters 140, 250 and 500 µm, with and without ad-ditional silicon oil. I used silicon oil rather than water as polystyrene beads arenot wettable by water with volume fractions ω = 0.01 and 0.04. The beads werepoured into the cup of the rheometer (50 mm diameter and 5mm bed depth) witha global packing fraction φ ' 0.63 (Fig. 2.3 (b)).

The rheometric equivalent of the differential pressure displacement curves are theso-called Lissajous curves, where one plots the stress as a function of the deforma-

2.4. Results 35

tion for a single oscillation cycle of the plate. In this set-up, we investigated theenergy dissipated in a Lissajous cycle which represents the resistance of the dry orwet granular media against the flow. Typical Lissajous characteristics are shownin Figs. 2.7 (b)-(d) for the dry and wet polystyrene beads [62].

This is observed that the results from shear cell and oscillatory test for differenttypes of flow, are in good agreement.

2.4 Results

2.4.1 Energy for flowing granular materials

The surprising conclusion from the shear cell experiments is that the wet sandflows more easily than the dry. This is evident from the pressure-displacementcurves shown Fig. 2.7 (a). The area enclosed by the pressure/strain curve isdirectly proportional to the work done, i.e., the energy dissipated during the cycle[44]. Therefore for the same overall deformation, the dissipated energy is muchsmaller for the wet than for the dry sand. Figure 2.5 shows σ versus γmax, thesemeasurements were taken for samples prepared with different packing fractions andliquid contents ω. Compaction of a granular assembly is the reduction of its freevolume when it is submitted to mechanical vibrations.

Recent studies have shown that the liquid bridges which determine the mechan-ical properties of granular systems, (dependent on the liquid content) exhibits amaximum in the range 0.01 ≤ w < 0.03. It was confirmed also by using X-raytomography that the onset for the coalescence of liquid bridges occurs around acritical water content of w ≈ 0.025 [19].

Figure 2.5 shows that for small deformations (γmax < 0.1), we found that dry sanddoes not resist any stress to within the accuracy of the experiment. On the otherhand, the wet sand behaves like a yield-stress fluid due to the liquid bridge network.For the almost completely dry and almost completely wet mixtures the yield-stressis quite low and the grains reach something resembling close packing when pouredinto the rheometer geometry, but for the intermediate liquid volume fractions theyield-stress is quite high and the material does not compact under its own weightgiving much lower grain filling fractions [26,63].

Accordingly, the inset in Fig. 2.5 shows the range γmax < 0.1, from which we geta mean value 〈σ〉 = (1.2± 0.4) kPa for the wet sand and 〈σ〉 = (1.2± 0.4) kPa forthe dry sand. For the wet sand this value keeps constant in the range of existenceof the liquid bridge network 0.01 ≤ ω ≤ 0.03; in good agreement with Ref. [64].In this paper Scheel et al. investigate the mechanical properties of the pile whichare largely independent of the amount of liquid over a wide range [65–68]. Theyresolved this puzzle with the help of X-ray microtomography and showed that theremarkable insensitivity of the mechanical properties to the liquid content is dueto the particular organization of the liquid in the pile into open structures.

Figure 2.6 (a) shows data for three characteristic mechanical parameters of a wetpile of the glass beads. The tensile strength, and the critical acceleration at which


Figure 2.5: Studies with the tube experiment, σ vs. γmax behaviour and correspondingfit for the dissipative flow range for dry sand with different packing fractions φ andwet sand (0.01 ≤ ω ≤ 0.03, φ = 0.63). The left inset shows the corresponding rangeγmax < 0.1. The right inset shows the characteristic energy E0 versus the packing fractionφ (triangles) compared with the wet sand (squares).

fluidization sets in are plotted as a function of liquid content, ω, in the range0 < ω ≤ 0.15. These results show that there is a sharp rise at very small ω, wherethe transition from the dry state to the wetted state (capillary bridges) takes place.Figure 2.6 (b) shows the liquid distribution and formation of the capillary bridgesbetween the grains. Actually the number of capillary bridges, the total number ofliquid clusters (defined as connected regions of liquid wetting more than two beads)and the liquid volume of the biggest cluster are shown as functions of the liquidcontent. Clearly, substantial changes in the morphology of the liquid structurestake place in the studied range. As ω is increased, the liquid structures merge intolarger clusters [64]. In Fig. 2.5 the onset of dissipative flow occurs for deformationslarger than about γonset ∼ 0.1 in all cases; this suggests that it is the size of thegrains rather than the presence or absence of the network that sets this strainscale. Further, σ grows more strongly for the dry than for the wet sand (Fig. 2.5),this means that one needs less energy to push the wet sand through the tube. Toquantify this, we note that beyond the yield point in Fig. 2.5, the data show anexponential dependence of the deformation on the stress:

γmaxγonset

= exp(σ

σ0). (2.7)

2.4. Results 37

Figure 2.6: Stiffness of a wet granular pile in comparison with the reorganization ofliquid structures. (a) Mechanical properties: tensile strength (filled squares), criticalacceleration for fluidization (open circles). The dashed horizontal line is a guide to theeye. (b) Images: sections through 3D tomograms of the analyzed samples at ω = 0.02, 0.04and 0.11, from left to right. Main panel: frequencies of liquid objects as extracted fromX-ray tomography data. Left axis (open symbols): average number of capillary bridgeson a sphere (triangles: by fluorescence microscopy and squares by X-ray tomography).Right axis: average number of clusters per sphere (filled symbols), and normalized volumeof the largest cluster (half-filled symbols) [64].

A fit with Eq. 2.7, applied, for example, for dry and wet sand with φ = 0.63,yields a constant σ = 55 and 14.7 kPa and onset values γonset = 0.1 and 0.14,respectively; with ν the volume of a grain, we define E0 = σν and E0 = σ0ν, so itis possible to rewrite Eq. 2.7 as:

γmaxγonset

= exp(E

E0), (2.8)

where E0 is an intensive property of the granular assembly, which in a mean-fieldpicture includes all the interactions and possible configurations of the grains [69].From the fits we then find E0 = 88 and 23 nJ for the dry and wet granulate withφ = 0.63, respectively. A mean-field meaning, all such interactions between re-gions, are subsumed into an effective “noise temperature” E0 = kBΘ [69–72]. Thisis based on the intuition that every yield event elsewhere in the material causes akick locally via the associated stress redistribution, and that many such kicks addup to an effectively thermal noise. This is assumed to create a new local equilibriumconfiguration with correspondingly new yield energy. We expect these activatedyield processes to arise primarily by coupling to structural rearrangements else-where in the system. In a mean-field spirit, all such interactions between regions


are subsumed into an effective noise temperature. The noise temperature, pro-posed phenomenologically to control the dynamics of a set of degrees of freedom.This parameter can be interpreted as a genuine non-equilibrium thermodynamictemperature, governing the degrees of freedom whose dynamics causes the systemto move among its various inherent structures or energy minima [69,71,73].

It is difficult to estimate the value of Θ for dry sand since the energy loss is sensi-tive to, for example, microscopic roughness and topology. For increasing packingfractione, we measure φ a range of Θ until around 6 PK (peta Kelvin) as shown inthe inset of Fig. 2.5, which is in agreement with previous measurements [70–72].

For the wet sand, Θ can be related directly to the liquid bridges. By consideringthe coordination number n = 6, the characteristic energy per bridge is E0/n =(4±2) nJ, which is in agreement with an estimate for the energy loss during bridgeformation and rupture ∆Ecap < πγωd

2/2 = 2.4 nJ [60,63].

2.4.2 Energy dissipated in wet and dry granular matter

As I explened in the prior sections of this chapter, the non-linear behavior ofmaterials in dynamic oscillations is characterized by a waveform of the outputsignal and this response of the material can be decomposed in a Fourier series. ALissajous figure can be used to represent the variation of the stress versus strainin the oscillatory shear test. So the term Lissajous curve denotes the projection ofthe oscillatory response curves onto the stress σ(t) versus strain γ(t) plane.

The energy dissipated per unit volume in a single cycle of Lissajous curves, can bevisualized by the area enclosed by the Lissajous curves [44]:

Ed =

∮σdγ = πG′′γ20 , (2.9)

For a linear viscoelastic material, G′′ is independent of γ0 and the dissipated en-ergy is a quadratic function of the strain amplitude. In this case, the Lissajousfigure is an ellipse. The non-linear viscoelastic behavior is also characterized by adeparture from the elliptic form and the distortions increasing with strain ampli-tude. Contrary to a linear viscoelastic material, the energy versus strain amplitudeis not a quadratic function. This curve for the perfect plastic materials is alwaysa rectangle with a high dissipated energy in comparison with the purely elasticmaterials where Ed goes to zero [49].

Equivalently, in the shear cell measurements when pushing sand through the tube,one can calculate the energy dissipated in a single cycle as the area of the differential-pressure curve loop. Figure 2.7 (a) shows the energy dissipated in this experimentfor dry and wet sand with liquid content ω = 0.03.

Figures 2.7 (b)-(d) also present the stress vs. strain plots for dry and wet sandto investigate material hysteretic properties in viscoelastic solid mechanics. Thesemeasurements have been carried out for 500, 250 and 140 µm beads’ diameter anddifferent liquid content. Recall that the energy lost by viscous dissipation per unit

2.4. Results 39

Figure 2.7: (a) Differential pressure curves for dry and wet sand measured with theset-up of Fig. 2.3 (a). In blocks (b)(d) Lissajous characteristics are shown for polystyrenebeads with and without silicon oil under confinement measured with the standard rheo-metric set-up of Fig. 2.3 for bead diameters (b) 250, (c) 500, and (d) 140µm. The liquidcontent ω indicates the wet loop. The strain amplitude γmax is indicated in (a) and (b).

volume is given by Lissajous curves and this energy represents the amount of workrequired to deform a material. So we need more energy to push the dry sand fordifferent type of materials and different conditions (i. e. liquid volume fraction ingranular system).

By calculating the area enclosed by these Lissajous curves in Fig. 2.7, for two verydifferent setups; shear cell and oscillatory test, we found that the dissipated energyfor the wet sand is smaller than that of the dry and it means that wet sand feelsless resistance aginst flow and consequently flows more easily. Physically, thesehysteresis, or energy dissipation, is supposed due to the inter-granular frictionduring rearrangement of the grains, which is expected to depend on the strainamplitude and the confining pressure which influences the contact forces. Thevalue of energy dissipated is strictly related to the nonlinear viscoelastic responses.

So we have examine the dissipated energy which is clearly related to the breakup


Figure 2.8: Dissipated energy per unit volume Ed versus strain amplitude γmax: (a) forthe range γmax > 0.1, calculated from the area of the differential pressure curve loop; and(b) integrating Ed from Lissajous loops for polystyrene beads. In both figures, squaresrepresent wet sand, and triangles dry sand. The insets show the same graphs in linearscale.

of the suspension structure. A comparison of two methods of experiments for dryand wet granular materials show that the resistance against flow for wet sand ismuch smaller than the dry sand and the cohesion arises from capillary force inthe wet matter describe the behavior of these systems. The bridges between thegrain create a network of grains that connected to each other by pendular bridges.These bridges leads to a force of attraction between the grains which is absent indry granular materials and hold the granular system together. In this case thisattractive force reduce the tendency of the system to dilate under flow and hencedecreases the flow resistance of wet systems. So we note that the Lissajous curveswould correspond to intersections and dissipative nature of the materials.

Surprisingly, we found that these results show that this conclusion is general fordifferent materials and does not depend on the size of the beads and the liquidcontent of the wet sand.

Figure 2.8 shows the energy for deformations beyond the yield point for bothexperiments. The comparison of Figs. 2.8 (a) and (b) shows that the results arevery similar qualitatively, but that a quantitative difference occurs in the measureddissipated energy. The differences in order of magnitude for Ed between the twoexperiments can be understood from both the differences between the two set-upsand those between the granular media.

2.5. Conclusion 41

2.5 Conclusion

The tube experiment is conceived in such a way that the deformation remainshomogeneous throughout the sample. In the rheology set-up, on the other hand, forthe larger deformations there is undoubtedly shear localization (banding) which ingeneral happens because this is the easiest way for the system to deform, that is, thedeformation that minimizes the dissipated energy. On the other hand, additionalexperiments for wet sand show that the opening of the loop σ varies inversely withthe diameter of the beads and also depends on the surface tension. Thus, althoughthe results from the two set-ups can only be compared in a qualitative manner,they do confirm the main result that the overall energy dissipation is smaller forwet than for dry sand.

The question remains then, why is this so? One possible explanation is the follow-ing: the sand should behave as a frictional granular system in the regime beyondyielding. When granular shear flows is prevented by confinement, shear is insteadaccompanied by normal forces. It is well known that for such systems, the shearstress under flow is directly proportional to the normal stress, i.e., the confiningpressure. This is significant evidence supporting the significance of dilation andnormal forces.

All the experiments have been carried out for the liquid volume fraction rangedbetween 0.01 and 0.04 which show the partially saturated regime of wetting. Theliquid leads to the formation of capillary bridges at the contact points betweenthe grains, and the surface energy (the curvature of the liquid interface) leads tocapillary pressure causing a force of attraction between the grains which is absentin dry granular materials. The bridges which create a network of grains connectedby pendular bridges are an adhesive force that hold the granular system together.In this case the suction force which is drawn back into the liquid under capillaryaction, should reduce the tendency of the system to dilate under flow, thus reducingthe confining pressure.

In the tube experiment a measure of the confining pressure is the mean value ofthe sum of the pressures at maximum shear, p = (|p1 + p2|−γmax + |p1 + p2|γmax)/2is shown in Fig. 2.9 (a). The advantage of the rheometric set-up is that it providesus with a direct measurement of the confining pressure. In the rheology exper-iments, which were carried out with a fixed gap of the cup-plate geometry, theconfining pressure is directly the normal stress measured by the rheometer duringthe deformation.

We have investigated the results by study the behavior of normal stress for dryand wet sand to see the frictional behavior in these systems. Figure 2.9 (b) showsthe measured normal stress as a function of the deformation for dry and wet sand.It is evident that the normal stresses are much larger for dry sand: the dry sandis much more dilatant and this provides a direct explanation of why the wet sandflows more easily than the dry sand. The origin of dilatancy when granular matterare sheared is that each particle needs to find room in order for the system to flow,and the system thus expands. But in the wet granular matter the liquid bridges


Figure 2.9: (a) Confinement pressure versus strain amplitude for the tube experiment.(b) Normal stress versus strain amplitude for the cup-plate rheometric set-up.

apply an adhesive force that hold the grains together and it redudce the normalforce or dilatancy in these systems.

Both dry and wet systems are expected to have a friction coefficient that is constantand somewhat smaller than unity. Since for frictional systems, the friction coeffi-cient is nothing more than the ratio of normal to shear forces, a smaller normalstress immediately implies a smaller shear stress, in line with the results presentedabove.

So, we have found that it is much easier to push wet sand than dry granular matterin a Poiseuille-like profile through a tube. Even if the capillary forces increase theyield-stress, the water promotes cluster formation and reduces effective intergrainfriction, whereas for dry sand the yield-stress is zero and a pure frictional behaviouris observed.

3.

How to construct the perfectsandcastle

Abstract: Just a bit of water enables one to turn a pile of dry sand into a spectac-ular sandcastle. Too much water, however, will destabilize the material, as is seenin landslides. Here, we investigate the stability of wet sand columns to account forthe maximum height of sandcastles. We find that the columns become unstableto elastic buckling under their own weight. This estimates the maximum height ofthe sand column; it is found to increase as the 2/3 power of the base radius of thecolumn. Measuring the elastic modulus of the wet sand, we find that the optimumstrength is achieved at a very low liquid volume fraction, less than 5%. Knowingthe modulus, we can quantitatively account for the measured sandcastle height.

3.1 Capillary bridges in wet granular material

The mechanical properties of granular matter are affected by the addition of liquidand these properties, make it an important phenomenon in industrial processes.The formation of capillary bridges between sand grains are the cause of the stiffnessof sculptured wet sand in a sand castle, as opposed to dry sand which can hardly ornot support its own weight [18,74]. Qualitatively, the liquid leads to the formationof capillary bridges at the contact points between the grains, and the surface energy(the curvature of the liquid interface) leads to capillary pressure causing a forceof attraction between the grains which is absent in dry granular materials. Whenliquid is added to granular matter, a three-phase system is formed, composed ofliquid, solid and interstitial fluid. Interfaces exist between liquid and solid, andliquid and displaced interstitial fluid (Fig. 3.1). The cohesive strength betweengrains increases sharply for very small volumes of liquid [75]. After this increase,however, the cohesivity is effectively constant until the liquid saturates the spacebetween the grains and cohesion decreases to zero, just as when the grains are

43

44 Chapter 3. How to construct the perfect sandcastle

Figure 3.1: The liquid bridges between beads as seen in fluorescence microscopy. Fromthe upper left corner and clockwise, are volume fractions of 0.1%, 0.3%, 3% and 6%,respectively. At a volume fraction of 0.1% the fluorescence of the fluid caught in thesurface roughness is seen to dominate that of the fluid in the bridges. At a volumefraction of 6% a significant number of the bridges are merged into large aggregates. Thebridges consist of water with added fluorescein and the interstitial air has been replacedby an oil with an index of refraction matching that of the glass beads [5].

completely dry [18,67]. Fluorescence microscopy can show the optical micrographsof the grains and capillary bridges between them, for different volume fractions(Fig. 3.1) [18, 26, 76]. More information and details about the different regimes ofwe granular materials can be found in Section 1.4.2.

These capillary bridges between the grains create a network of grains connectedby pendular bridges and allows, for example, creating complex structures such assandcastles. Not many quantitative studies on the mechanical properties of wetsand exist, despite the fact that the handling and flow of granular materials isresponsible for roughly 10% of the world energy consumption [13].

Since in many cases the humidity in the air is sufficient for liquid bridges to formbetween sand grains, one would expect the mechanical behaviour to be well known.This is not the case, in spite of the fact that the stability of wet granular packing isof paramount importance for civil engineering purposes and that the adhesive forcesdue to the presence of liquid bridges are also extremely important in agriculture[56], and geophysical applications (i.e., soil stability), of which sandcastles aremerely an unusual example [59, 64, 77–80]. For sandcastles, the only estimate in

3.2. What is buckling? 45

the literature, argues that the stability is related to the capillary rise in the granularmedium, and arrives at a maximum sandcastle height of roughly 20 cm [81]. Thisis in stark disagreement with the observation of sandcastles several metres high,and the common observation that stability depends on the base radius of the sandstructure. To account for the (in)stability of sandcastles, we show here that it issufficient to consider that the limit of instability is reached when a column of sandundergoes a buckling transition under its own weight when exceeding a criticalheight [75].

3.2 What is buckling?

When a slender structure is loaded in compression, for small loads it deformswithout any noticeable change in geometry. On a critical load value, the structuresuddenly deforms and may lose its ability to carry the load and buckles [82,83]. Thedesign of structures is often based on strength and stiffness considerations. Strengthis defined to be the ability of the structure to bear the applied load, while stiffnessis the resistance to deformation. Therefore, buckling is an important considerationin structural design, especially when the structure is slender and lightweight.

Buckling, also known as structural instability, may be classified into two categories:bifurcation buckling and limit load buckling. In bifurcation buckling, the deflectionunder compressive load changes from one direction to a different direction (e.g.,from axial shortening to lateral deflection). The load at which the bifurcationoccurs in the load-deflection space is called the critical buckling load. In limit loadbuckling, the structure attains a maximum load without any previous bifurcation,i.e., with only a single mode of deflection [82].

3.2.1 Historical review

The first study on elastic stability is attributed to Leonhard Euler (1707 − 1783),who used the theory of calculus of variations to obtain the equilibrium equationand buckling load of a compressed elastic column. This work was published in the“appendix De curvis elasticis” of his book1.

After developing the energy approach to solve the mechanics problems by Lagrange(1736 − 1813), and the concept of stability in the 19th century, Karmen (1881 −1963) began his work on the elastic buckling of columns. Also in 1881 Greenhillinvestigated the buckling of a heavy column and estimated the maximum height as:Hmax = (7.83ρ/E)1/3, where ρ and E are the density and rigidity of the column,respectively. This length sets the maximum height for trees if we assume the treesare prismatic and the branches are neglected.

Nowadays the stability of the structures is an important problem for civil engi-neering and soil mechanics. In addition the study of the stability of wet granular

1Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes.


Figure 3.2: The world’s tallest sandcastle was built on Myrtle Beach in South Carolinain 2007 Sun Fun Festival. The height of this structure was 15.1 m [84].

materials is of practical interests in constructing the perfect sandcastles. Sandsculpting as an art has become very popular in recent years and since 1989 a worldcompetitions has been held in coastal areas and hundreds of annual competitionsare held all over the world. Figure (3.2) shows one of these sand sculptures.

3.3 A mechanism for buckling

Buckling is one of the important examples of bifurcation that provides models oftransitions and instabilities. If a load is placed on a beam, the beam can supportthe load. But if the load is too heavy, the beam becomes unstable and buckles(Fig. 3.3).

In the Landau theory of phase transitions, the potential of a system as the form ofa power series in the order parameter x is given by:

3.3. A mechanism for buckling 47

Figure 3.3: A schematic of buckling phenomenon.

Figure 3.4: A phenomenological explanation of buckling in one dimension. When aload is less than the critical load, the phenomenological potential has a stable equilibriumat zero. When the load exceeds the critical value, the stable equilibrium changes to anunstable point and two stable points appear.

V (x) = λ0 + λ1x+ λ2x2 + λ3x

3 + λ4x4 + .... (3.1)

and the quantity x takes the small values near the transition point. In the absenceof extra load, the underlying Hamiltonian is invariant to x translation. It mustthen follow that V is also invariant to the sign of the reflection x. Hence, allodd coefficients of x must vanish. For qualitative purposes, the potential may betruncated to the fourth order, as:

V (x) = αx4 − βx2 (3.2)

An equilibrium is defined to be stable if all small disturbances away from it dampout in time and it is unstable when disturbances grow in time. Stable and unstableequilibriums are represented by stable and unstable fixed points, respectively. To


Β

X fp

Figure 3.5: Bifurcation diagrams. Location of stable (solid line) and unstable (dashedline) equilibrium vs. external load. At the critical point; βc = 0 the stable equilibriumchanges to an unstable point and two stable points appear.

find the fixed points, we should set V (x) = 0 and to determine stability, we plot itas a function of x:

V (x) = 4αx3 − 2βx = 0 → x = 0, x2 =β

2α. (3.3)

By considering α > 0, β is a parameter which may be positive or negative. Whereβ > 0 there are three fixed points, two stable and one unstable equilibrium atx = 0. For β < 0 the local minimum at x = 0 corresponds to stable fixed point(Fig. 3.4).

β > 0→ x = 0, x = ±√

β

2αβ < 0→ x = 0 (3.4)

In the buckling problem, β corresponds to the extra load β ∼ F − Fc. Figure 3.5shows the equilibrium points as a function of the load parameter (β).

3.4 Buckling of a column under its own weight

Consider a perfectly straight, uniform, homogeneous column of rigidity E, lengthL and density ρ. Figure 3.6 shows the forces and moments acting on a crosssection of a buckled column. The moment-displacement relation according to theEuler-Bernoulli beam theory that describes the relationship between the beam’sdeflection and the applied load, is given by:

M = −EI d2w

dx2, (3.5)

3.4. Buckling of a column under its own weight 49

Figure 3.6: Heavy column under its own weight.

where w is the transverse displacement of the buckled column and x is the longi-tudinal coordinate. A moment balance yields the linearized equation:

dM + ρ(L− x)θdx = 0. (3.6)

By substituting this equation in Eq. (3.6), the governing equation is given by:

EId2θ

dx2+ ρ(L− x)θ = 0, (3.7)

which displacement w w is related to the slope θ by:

θ =dw

dx. (3.8)

Let

β =ρL3

EI, s =

x

L, (3.9)

where s is between 0 and 1. Then Eq. (3.7) becomes:

d2θ

ds2+ β(1− s)θ = 0. (3.10)

By considering β > 0 and r = (1− s)|β|1/3, Eq. (3.10) further simplifies to:

d2θ

dr2+ rθ = 0. (3.11)

The general solution of this equation is given by the Airy functions:


θ = C1Ai(r) + C2Bi(r). (3.12)

The Airy functions Ai and Bi are related to the Bessel functions by this relation:z = 2

3r3/2, then:

Ai(−r) =

√r

3[J1/3(z) + J−1/3(z)], A′i(−r) =

r

3[J2/3(z)− J−2/3(z)](3.13)

Bi(−r) =

√r

3[−J1/3(z) + J−1/3(z)], B′i(−r) =

r√3

[J2/3(z) + J−2/3(z)].(3.14)

Applying the boundary conditions at the top and bottom ends, one can see thatthe buckling solution depends on the Bessel function of the first kind of order −1/3.

So an elastic rod becomes elastically unstable and buckles under its own weightwhen exceeding a critical height hcrit as:

hcrit = (9J−1/3

16

GR2

ρg)1/3, (3.15)

where G is the elastic modulus, R the column radius, ρ is density and g the gravi-tational acceleration [82,83,85].

3.5 The shear modulus of wet granular materials

To be able to calculate the maximum height of a sandcastle, we need to quantify theshear modulus. Also one of the most spectacular properties of granular materials ishow the network of liquid between the grains changes the strength and macroscopicproperties of these materials. To express the strength of the materials and give ananswer to the question of what is the optimum liquid volume fraction, we shouldmeasure the elastic shear modulus. Shear modulus as one of the quantities formeasuring the stiffness of materials, is relates the tensile strain to an applied tensilestress [86].

To understand the dependency of the strength of the grains to the volume fraction,we need to know what happens when one adds a small amount of fluid to a stackof spherical grains. Møller et al. have shown that the macroscopic strength ofdifferent wet granular materials depends with a power of 2/3 on the microscopicelastic modulus of the individual grains, with a power of −1/3 on the radius, andwith a power of 1/3 on the surface tension [26]. They have explained it by assumingthat the attractive capillary force between two grains deforms the grains elastically,yielding a “spring constant” for further deformation.

The surface tension of the fluid pulls it into small bridges connecting individualgrains. According to Section (1.6), the capillary force between two grains is givenby:

Fcap = −πr21∆p+ 2πr1γ = −πr21Cγ + 2πr1γ, (3.16)

3.5. The shear modulus of wet granular materials 51

Figure 3.7: Schematic of a liquid bridge between two beads.

where γ is the surfaces tension of the liquid and r1 and C are the radius and thecurvature of the bridge respectively. Figure 3.7 shows the schematic of a liquidbridge. Curvature C is a function of a, r1 and the distance between the beads, d.In a case in which d = 0 and r1 a one can approximate C ≈ −1/r ≈ −2a/r21 sothat the capillary force is 2πaγ, that is very good approximation at the intermediatevolume fraction. The factor f(Vf ) that is between 0 and 1, shows the dependencyof the capillary force to the volume fraction:

Fcap = 2πaγf(Vf ). (3.17)

The attractive capillary force, due to the liquid bridge between two beads, sucksthe beads together. If this attractive force removes the distance between two beads,it can deform them elastically until the capillary force is balances by the elasticresponse of the beads. Classical contact mechanics is associated with HeinrichHertz in 1882, is the basis for contact mechanics even today. He solved the prob-lem of contact between two elastic bodies with curved surfaces. According to theHertzian response theory, when two identical spherical beads contact each otherwith compression δ, the relation between the elastic normal force FE and δ is givenby:

FE =4√

2

9a1/2Eδ3/2, (3.18)

where E is the Young modulus and R is the radius of the spheres. Linearizingthe total force around the equilibrium position where Fcap = FE gives a springconstant against further displacement of the beads:

Ftotal = 0→ δeq = (9πa1/2γf(Vf )

2√

2E). (3.19)

After a further displacement by ∆δ the total force given by:


F =4√

2

9a1/2E((δeq + ∆δ)3/2 − δ3/2eq ) ≈ 2

√2

3a1/2E2δ1/2eq ∆δ ≈ (3.20)

≈ (8π

3a2Eγf(Vf ))1/3∆δ ≡ k∆δ.

The macroscopic shear modulus G of a macroscopic cube of dimension L containinga large amount of grains can be defined as the ratio of stress and strain:

E = 2(1 + υ)G ∼ F/L2

∆x/L, (3.21)

where ∆x/L is the strain, F/L2 the stress and υ ≈ 0.5 the Poisson ratio. Assumingthat at the level of single particles the capillary and elastic forces are balancedfor each pair of grains and using the simple Hertz contact for the grain contactelasticity, the optimum strength G can be found by averaging over all pairs as:

G = 0.054a−1/3E2/3γ1/3f(Vf )1/3, (3.22)

where a is the radius of the grains, E is the Youngs modulus of the grain materialand γ is the surface tension of the liquid-air interface, respectively.

3.6 Buckling of the sand columns

Capillary forces significantly affect the stability of granular matters. So he forma-tion of liquid bridges between sand grains are the cause of the stiffness of sculpturedwet sand in a sand castle. In this chapter we analyze the stability of sand columns(wet granular materials) and find that the maximum height of a column become un-stable to elastic buckling under its own weight when exceeding this critical height.

In the previous sections I explained about the analytical solution of hcrit for acylindrical column and also the macroscopic shear modulus G to find the optimumstrength of this system. To investigate the analytical results, I present the exper-iments of the stability of sand columns and find a recipe to construct a perfectsandcastle with the optimum strength.

3.6.1 Experimental techniques

To verify the analytical solution for maximum height of a sand column, experimen-tally, beach sand with an average radius of 100 µm was mixed with a small amountof deionized water.

Cylindrical “sandcastles” were constructed using non-wetting PVC pipes of dif-ferent diameters cut in half over the length of the tube. The two halves wereassembled, and the wet sand was put in the tube standing vertically on a surface.

3.7. Results 53

The wet sand was poured into the pipe in small portions and compacted by drop-ping a thumper into the pipe at least 70 times. This process was repeated until thepipe was filled with sand up to a certain height. The two halves of the cylindricaltube were then carefully removed and if the sand column was stable, a new exper-iment was launched filling the tube to a larger height, until the column collapsedand could not support its weight.

So several experiments were carried out to measure how tall and thin sand columnswe could make before they fall down. We observed that for a special height, thesand column were stable for a long time, but a taller column with the same radiusbuckles after separating the pipes. Several experiments were undertaken at eachfilling height to ensure the reproducibility of the results. These experiments andthe collapsed sand columns showed that in a critical height, the column (with thecertain radius) cannot support its own weight and buckles suddenly.

Figure 3.8 shows two columns of sand with height 27 cm and 60 cm correspondingto radii R = 2 cm and 7 cm. This procedure was followed for 8 pipes of radiiranging between 0.5 and 7.5 cm.

3.7 Results

To account for the (in)stability of sandcastles, we show here that it is sufficient toconsider that the limit of instability is reached when a column of sand undergoesa buckling transition under its own weight. An elastic rod becomes elasticallyunstable and buckles under its own weight when exceeding a critical height hcritwhich for a cylindrical column is given by [27]:

hmax = (9J2−1/3

16

GR2

ρg)1/3, (3.23)

whereG is the elastic modulus, R the column radius, ρ is density, g the gravitationalacceleration, and J ≈ 1.86 is the smallest positive root of the Bessel function of thefirst kind of order −1/3. From these arguments, the maximum height varies withthe base diameter as hmax ∼ R2/3. The experimental data for the maximum heightas a function of the column radius compare rather favorably with the theoreticalexpression for buckling (Fig. 3.9). The exponent of experimental data is in goodagreement with the theory:

hexp ∼ R0.7±0.05 (3.24)

To be able to quantitatively compare the buckling prediction for the maximumheight of a sandcastle, we need to quantify the shear modulus. A recently in-troduced model for the strength of wet granular matter presented in Section 3.5.Assumes that when one adds a volume of liquid to grains, the capillary attractiveforce and elastic response from the Hertz contact between two spheres will be bal-anced. As two beads are always separated by at least the surface roughness, below acritical liquid volume fraction of about 0.2%, the bridges between the beads cannot


Figure 3.8: Sandcastles with diameters 2 cm and 7 cm.

form. At higher volume fractions, the bridge force is dominated by the curvatureof the meniscus and at even higher volume fractions the bridges start to merge intolarger pockets of fluid. According to the Eq. 3.22, the macroscopic shear modulusG can be formed as:

G = αa−1/3E2/3γ1/3, (3.25)

where α is the constant of proportionality that I explaned and calculated it in thesection 3.5. This constant parameter expresses how much the individual capillarysphere-sphere bonds are deformed relative to the globally imposed strain. To com-pute an estimate of α by this model, we took a simple cubic crystal of frictionlessspheres which suck each other because of the capillary force between them. Thesebeads deform elastically until the attractive capillary force balanced with the elas-tic response of the beads. Average over all sphere-sphere bond orientations givesα ≈ 0.054.

Figure 3.10 shows the measured elastic shear modulus G as a function of the liquidvolume fraction for different granular materials. These results show that wet sandhas a shear modulus similar to that of spherical glass beads, while that of spherical

3.8. Conclusion 55

0.1 1 1010

100

Radius (cm)

h max

(cm

)

Figure 3.9: Experimental data points and theoretical prediction of the maximum heightof a sandcastle as a function of its radius. The solid line is the theory without anyadjustable parameters using G = 0.054a−1/3E2/3γ1/3, where a = 100 µm, E = 30 GPa,γ = 70 mN/m. The density of the sand is ρ = 2.6 gr/cm3. The small but systematicdiscrepancy between the theory and the experiments is likely to be due to perturbationsthat arise when we remove the PVC pipes, leading to a somewhat smaller maximumheight than the theoretical one.

polystyrene beads is much smaller. In addition the optimum strength is achievedat a very low liquid volume fraction of 1− 3% [26].

Also to predict how the modulus of a material depends on the bead material, thebead size, the fluid surface tension, and the liquid volume fraction one needs topredict how f(Vf ) varies. Figure 3.11 shows the elastic moduli of the materialsrescaled by G → G/(0.054R1/3E2/3γ1/3) (both the experimental data and theprediction) or the shape of f(Vf )1/3. By this model and experimental data wetake the function f(Vf ) ≈ 1 to determine the maximum strength; f(Vf ) shows thedependence of the elastic modulus on the liquid volume fraction and is unity forthe optimum volume fraction.

3.8 Conclusion

This model gives a very accurate result for the maximum strength of the sandpacking, which makes it possible to compute how high sandcastles can be builtfrom the predicted elastic modulus for any size grains of wet sand. Using typical


Figure 3.10: Measured elastic shear modulus G as a function of the liquid volumefraction for different granular materials; 1 : 100 µm polystyrene beads and silicone oil;2 : 100 µm glass beads and water; 3 : 100 µm glass beads and silicone oil; 4 : 100 µm sandand water; 5 : 25 µm glass beads and silicone oil; 6 : 3 µm PMMA beads and silicone oil;7 : 100 µm polystyrene beads and silicone oil in a larger-gap vane-in-cup geometry [26].

values for beach sand, a cylinder with a radius of 20 cm for instance could be as tallas about 2.5 m, which is in quite good agreement with that which can be observedfor real sandcastles [87]. This estimate is an immense improvement compared toa previous dimensional analysis result which gives roughly 20 cm as the maximumheight of a sandcastle, independent of the base diameter [81]. For our cylindricalsandcastles, using the optimum strength in the buckling arguments, we arrive ata quantitative theory for sandcastle stability that agrees with the measurements(Fig. 3.9).

Can we use these new insights to build taller sandcastles? From hmax we see thatbesides the sandcastle diameter, the most potent power is associated with α/ρg.We cannot change g, but α can be increased by compaction which is always carriedout by sandcastle builders. Also, we could decrease the effective density, ρ, of oursandcastle by plunging it under water. For normal sand however, this will destroythe liquid bridges between grains and thus the strength of the material. However, ifhydrophobic sand is used the roles of water and air interchange completely [88]. Inthis case the air and not the water “wets” the grains and we can simply interchangewater and air, which does not change the bridge force, since the force betweenbeads remain constant, but the effective density of compacted sand changes from1.6 g/cm3 to 0.6 g/cm3 when immersed in water. This makes it possible to buildunderwater sandcastles, which are even more spectacular than normal ones.

To construct this underwater sandcastle, commercial hydrophobic sand was used.The presence of this hydrophobic compound causes the grains of sand to adhere

3.8. Conclusion 57

Figure 3.11: The rescaled elastic modulus of different materials which shows the behviorof f(Vf )1/3 (see the caption of Fig. 3.10 for the legends) [26]. Inset: distribution of thesize of the grains in the sandcastle experiment [75].

to one another to minimize the surface area when exposed to water. The differentelements of this sandcastle were moulded under water, saturated with interstitialair.

Cylindrical parts were made using a plastic pipe cut into two halves, semi-cylinders,and making them into a hinge cylinder. The dry sand was poured into the pipeand it was submerged in to the water. The air between the grains in the sandcylinder was sucked out by a syringe and then the pipe was removed. The top ofthe castle was constructed by a pile of sand under water shaped by hand. Aftermodelling, a syringe was used to suck out air from the elements, reducing the fluidvolume fraction from about 40% to about 10% in order to increase the strength ofthe material before simply moving them into place by hand.

These results are of practical interest for civil engineering and soil mechanics, aswell as being of fundamental interest to achieve a better understanding of partiallysaturated granular materials. In addition, it explains the maximum height of, andprovides us with a recipe to construct the perfect sandcastle.


Figure 3.12: Using commercially available hydrophobic sand it is possible to build anunderwater sandcastle. Since the force between beads remains constant, but the effectiveweight of the sandcastle is reduced by a factor of 3, it is possible to build more spectacularsandcastles underwater than above.

Part II

Rheology of Wet Granular Ma-terials

59

4.

Flow irreversibility ingranular suspensions

Abstract: Slow, viscous flows are perfectly time-reversible. However, recent ex-periments have shown that if a particle paste is subjected to slow oscillatory shear,the particle motion becomes irreversible above a certain critical deformation, theorigin of the irreversibility being unclear. In this chapter by Magnetic ResonanceImaging to measure the homogeneity of the suspensions we show an irreversiblemigration of particles from high-into low shear rate regions. Second, we find that,above a critical deformation large normal forces appear in the suspensions pointingto frictional contacts between the particles. Such contacts also lead to irreversibil-ity in the motion of the particles, and in addition give a quantitative criterion forthe onset of irreversibility.

4.1 Introduction

One of the famous experiments of G. I. Taylor is the illustration of the reversibilityof Stokes flow. In the experiment, a drop of ink is injected inside a viscous fluidin the gap between two concentric cylinders. When the inner cylinder is rotatedseveral turns, the ink spot is no longer visible. However, when the same cylinderis rotated in reverse exactly the same number of turns, it reappears again. Also,theoretically, it is easily shown that the motion of an incompressible fluid at lowReynolds number, governed by the Stokes equation, is time-reversible.

When the suspended particles are small enough, the inertia of the fluid and particlesare unimportant in determining their dynamics. We refer to these systems asStokesian suspensions, as the fluid motion is governed by the Stokes equations.Unless the particles are very small, the effect of colloidal forces and Brownianmotion can be discounted. The Stokes equation is obtained from the Navier-Stokes

61

62 Chapter 4. Flow irreversibility in granular suspensions

equation for incompressible fluids simply by omitting the non-linear term and thetime derivative of the velocity field:

∇p+ η∇2u = 0

∇.u = 0 (4.1)

If we assume that that a velocity field u(r) is a solution of the equation, with acorresponding pressure field p(r), −u(r) will also be a solution provided only thatwe reverse the sign of the pressure gradients, as well as that of the velocities, atevery solid boundary. Eq. 4.1 is then again satisfied, since its two terms are replacedby their negatives and the boundary conditions are appropriately changed [5, 89].

One may therefore anticipate that, at a low Reynolds number, every incompressiblefluid should behave reversibly. It consequently came as a surprise that particlesin suspensions of non-Brownian rigid particles can fail to return to their initialposition under oscillatory shear [89, 90]. In the experiments of references [89, 90],rather concentrated suspensions were subjected to a large-amplitude oscillatoryshear; one would then naively expect that after every oscillation all particles re-trieve their initial position. However, in the experiments it was found that thereexists a maximum strain value γ0m, above which the particle motion is irreversible.Understanding the mechanism that leads to this irreversible motion is currentlythe subject of debate [89–95].

In this chapter, we investigate in detail what happens if the same non-Browniansuspension that was used in [89,90] is again subjected to an oscillatory flow. Theseexperiments unambiguously demonstrate that there is an irreversible migration ofparticles from high into low shear rate regions. We use Magnetic Resonance Imag-ing (MRI) [96] to measure the homogeneity of the suspensions. The experimentsusing by MRI techques have been carried out in collaboration with the StatisticalPhysics Laboratory in Paris.

Second, above a given strain γ0c ' γ0m value [90], we find that large normal forcesappear in the suspensions. Those forces are the signature of a transition betweena viscous regime, in which the particle contacts are lubricated, and a collisionalregime in which the contacts are frictional [97]. It is evident that frictional contactswill also lead to irreversibility in the motion of the particles [98]. In addition, theonset of frictional behaviour gives a criterion for the onset of irreversibility thatagrees reasonably with both the experiments and the numerical simulations of [90].

4.2 Magnetic resonance imaging

Magnetic resonance imaging (MRI) is a technique to visualize the structures of asample in detail and can provide nice colorful 3D pictures of it. Also NMR (nuclearmagnetic resonance) is a similar technique for obtaining more detailed chemical in-formation about molecules. It is known that most illustrations of the magneticresonance imaging technique occur in the medical field. But it has different ap-plications in chemical engineering, fluid mechanics and fluid transport in porous

4.3. Flow irreversibility in granular suspensions under large amplitude oscillatory shear63

media and these wide ranges of applications have led to five Nobel Prizes awardedto discoveries related to NMR or MRI.

MRI is a technique which uses the spin properties of atomic nuclei to obtain re-solved information about the structure and dynamics inside a sample. Considerthe magnetic moment M of the nucleus of a hydrogen atom associated with anorbiting nucleus lies along the same direction as the angular momentum L of itand proportional to it. So M = γL, where γ is the gyromagnefic ratio.

In MRI, the sample is immerged in a strong magnetic field ~B = B0 ~ez, where inthis case ~M reaches a stable equilibrium values, ~M = γ2~2 ~B0/kT and the averagemagnetic moment of the protons becomes aligned with the direction of the field.This situation can be tilted by a radio frequency magnetic field. The energy of themagnetic moment is given by E = −M.B and the torque on the magnetic momentwill be:

d ~M

dt= γ ~M × ~B0. (4.2)

So after tilting the magnetic moment M processes around the magnetic field B0 atthe Larmor precession frequency ω0 = γB0. This frequency, known as the resonancefrequency, is absorbed and flips the spin of the protons in the magnetic field.

In a MRI experiment (Fig. 4.1 (a)), the sample is placed inside a coil which ismounted between the pole pieces of a magnet. The magnet creates the constantand homogeneous magnetic field B0 inside the sample [96,99,100].

4.3 Flow irreversibility in granular suspensions un-der large amplitude oscillatory shear

Here , we investigate in detail what happens if a non-Brownian suspension subjectedto an oscillatory shear flow. I have carried out the experiments by parallel plategeometry to find The transition from reversible to irreversible behaviour. Theseresults show that above a critical deformation large normal forces appear in thesuspensions pointing to frictional contacts between the particles.

Also we use Magnetic Resonance Imaging (MRI) and Couette geometry to measurethe homogeneity of the suspensions. These experiments unambiguously demon-strate that there is an irreversible migration of particles from high into low shearrate regions.

4.3.1 Experimental techniques

We studied the rheological properties of isodense granular suspensions composedof non-Brownian spherical particles immersed in a Newtonian fluid. The granularmaterial was made of monodispersed polystyrene spheres (diameter d = 140± 5%,density 1050 kg.m−3) from Dynoseeds. We used a mixture of distilled water (19%in weight) and Triton (81% in weight) [90] as the interstitial fluid in order to get a


perfect density matching, as checked by centrifuging the samples: no sedimentationor creaming occurred.

The first studies and experiments on reversibility of non-Brownian suspensionshave been carried out by applying a sinusoidally strain with a fixed frequency ina parallel plate geometry (with diameter 50 mm). This geometry enables us tomeasure the normal forces appear in the suspensions pointing to frictional contactsbetween the particles. Such contacts could show the irreversibility in the motion ofthe particles, and give a quantitative criterion for the onset of irreversibility. Thesample with the liquid volume fraction ranged between 30% and 60% is subjectedto oscillatory shear γ(t) = γ0 sinωt at a fixed frequency 1 Hz.

Straining the system to strain amplitude γ0 evolves the system forward in time;by reversing the flow in the next quarter cycle, we can check to see if the particlesreturn to their initial positions.

Other experiments of classical rheology measurements were performed with Couettegeometry (inner cylinder radius Ri = 4.15 cm, outer cylinder R0 = 6 cm, height11 cm) on a classical rheometer that controls the stress or the strain (Fig. 4.1 (b)).The volume fraction was fixed at 40% for all of the experiments discussed here, asin [89,90]. In the following, we define that the strain amplitude corresponds to theratio of the maximum azimuthal translation of the inner cylinder and the size ofthe gap. it means that when the sample is subjected to oscillatory shear at a fixedfrequency ω and a fixed strain γ0, the time dependence of the strain γ0, is givenby γ(t) = γ0 sinωt. The strain amplitude γ0 ranged between 0.5 and 3.5 and thefrequency between 10 mHz and 1 Hz. Rough lateral surfaces are used in order toavoid wall slip effects. The order of magnitude of the roughness was the scale ofone grain diameter. The largest Reynolds number we reached was then around 1.

The error in the measurement of the volume fraction is around 0.25% for MRIexperiments, expected very close to the wall where the error is in the order of apercent.

4.4 Results

4.4.1 Viscosity and particle migration

Returning to the rheology experiments, during one experiment, the total accumu-lated strain experienced by the particles over a run is γ = 4γ0 × n, where n is thenumber of cycles (because the strain is γ0 in each quarter cycle) and in rescaledform is given by:

γ = 4γ0 × n× (R0 −Ri)/d, (4.3)

where d is the grain diameter. Performing oscillatory experiments at fixed strainand frequency, we observed a slow decrease of the complex viscosity (Figs. 4.2and 4.3). Returning to the complex representation of the oscillatory motion, as analternative to the complex shear modulus, we can define complex viscosity as:

η? =√G′2 +G′′2/ω. (4.4)

4.4. Results 65

Figure 4.1: Oscillatory experiments by using: (a) MRI technique to measure the localvolume fraction and (b) Couette geometry to measure the complex viscosity.

Similar observations had been made for suspensions under steady state flow longago [101], and were interpreted as shear-induced particle migration: the particleshave a tendency to move from high to low particle pressure regions due to theunequal frequency of collisions: if a shear rate gradient exists in the sample therewill, on average, be more collisions on the high shear rate side than on the low shearrate side of the particle, causing a slow drift towards the low shear rate side [101].Recently, a viscosity decrease, concomitant with a non-uniform volume fractioninside the gap of a Couette cell after shearing for a long time at a constant rate,was also observed directly using MRI [102]. Using the same MRI technique wecan also measure the local volume fraction in an oscillatory flow.

In the classical rheology experiments, the migration, evident as a slow decrease inthe measured viscosity, is very slow, necessitating total deformations of several tensof thousands [103–106]. Note that we did not observe a plateau for long periods oftimes (Figs. 4.2 (a) and 4.3 (a)), as expected, if only migration occurs. However,since the experiments were very slow, the latter was very likely due to sedimentationor creaming; we very carefully matched the density at a given temperature, butsmall changes in the temperature of the laboratory will cause a density mismatchthat is visible over very long times periods as a slow change in the viscosity.

Whether the apparent viscosity goes up or down depends on the precise geometryof the experimental setup; in general the apparent viscosity increases in time (adetailed investigation of a similar system was published recently [107]). Howeverif there is a dead volume into which the particles may cream or sediment, theviscosity may go down. The dead volume is present in our experiment; however we


Figure 4.2: The suspension is subjected to an oscillatory shear flow in Couette geometryfor different strain amplitudes γ0 from 0.05 to 3.5 at frequency f = 0.05 Hz. (a) Thepaste has a volume fraction φ = 40%. Time evolution of the complex viscosity, (b)Rescaled complex viscosity η? =

√G′2 +G′′2/ω as a function of the total deformation

γ = 4γ0 × n × (R0 − Ri)/d. The curves are rescaled by the asymptotic value for thehighest total deformation.

cannot rule out that the slow decrease is also partly due to slow migration. Morerecently, a second, and faster type of particle migration was observed in a systemsimilar to the one discussed here; the fast migration there was attributed to thenormal stresses [101–103].

Also we carried out the MRI experiments with the same Couette geometry as forthe rheology (Fig. 4.1 (b)). The inner cylinder was driven by a motor that allowedus to adjust the amplitude and frequency of the oscillation. We performed theoscillations outside of the scanner and subsequently gently introduce the Couettegeometry into the MRI apparatus to verify the homogeneity of the sample [96].The measured MRI signal is proportional to the density of protons. By choosinga suitable frequency, the MRI scanner only detects the protons inside the liquid(not inside the polystyrene beads). We were then able to measure the local volume

4.4. Results 67

Figure 4.3: The suspension is subjected to an oscillatory shear flow in a Couette ge-ometry for different strain amplitudes γ0 from 0.05 to 3.5 and frequencies. The pastehas a volume fraction φ = 40%. (a) Time evolution of the rescaled complex viscosity.(b) Rescaled complex viscosity η? =

√G′2 +G′′2/ω as a function of the total deformation

γ = 4γ0×n×(R0−Ri)/d. The curves are rescaled by the asymptotic value for the highesttotal deformation.

fraction of beads throughout the gap of the Couette cell.

The measured density profiles show that the particle concentration is homogeneousinitially, but after a long time of shearing the volume fractions is significantlylower close to the moving inner cylinder, and increases roughly linearly with thedistance from the axis (Fig. 4.4). This confirms the idea that the viscosity decreaseswith time inside the gap due to particle migration. The migration automaticallygenerates a local variation of the viscosity, from which the observed global decreaseof the viscosity results.

It is worth while stressing that the total deformation these samples have undergoneis strictly zero, showing directly that the migration process is irreversible, evenfor lower strains. This is therefore a possible explanation for the observation ofirreversibility in the previous experiments.


Figure 4.4: Density profile of the granular suspension (volume fraction φ = 40%) in thegap of the MRI Couette cell at different time. The open symbols correspond to the initialdensity profile and the dashed line is a guide for the eye. Circles are taken at t = 30minutes and square at t = 60 minutes. The frequency is 51.5 mHz and the amplitude ofthe deformation is fixed at γ0 = 15.

We found that the amount of migration and hence the value of the complex vis-cosity depends only on the total strain (see Figs. 4.2 (b) and 4.3 (b)). For differentoscillation frequencies, the viscosity values collapse on to a single curve when plot-ted as a function of the strain. This means that migration even occurs for the loweststrain amplitudes. In Figs. 4.2 (b) and 4.3 (b), the complex viscosity curves arerescaled by the asymptotic value for the highest total deformation; as already men-tioned, the observed small differences are likely to be due to residual sedimentationor creaming effects [107]. The conclusion from these experiments is that even forsmall deformations under γ0, the shear-induced diffusion results in an irreversibleflow for long-time scales. It is worthwhile underlining that the migration happensin the radial direction, whereas the particle trajectories observed in [89, 90] weredetermined in the plane perpendicular to this direction; thus we cannot directlycompare the two data sets.

In addition, although it may very well be that for small amplitude oscillations, thetime over which the migration occurs becomes so large that in practice it is notvisible, this does not account for the relatively well defined critical strain beyondwhich the flows were irreversible in the earlier experiments [89, 90]. There shouldbe another mechanism that probably acts simultaneously with the migration.

4.4. Results 69

4.4.2 Critical strain for irreversibility

The transition from reversible to irreversible behaviour is fundamental to statisticalphysics.

! " !" " " " "

!

! "

! #

! $

! %

! &

! " !" " " " "

"

#

$

%

!"

#

!"#$%&'!"()&* +

$%&'()# *"

+,-./0#1&'2%(,)

#34*

#54*

#64*

#66*

#74*

'"

+,-./0#1&'2%(,)

#34*

#54*

#64*

#66*

#74*

&

#

,-)$"&,.")//&*0$+

$%&'()# *"

! " #! #" "! "" $!

!

"!

%!!

%"!

&!!

!"#$#%&' (

$"&#) *+

,

!"#$%& '()*+,"- ./0

Figure 4.5: (a) Shear stress measurements as a function of the applied strain γ0, (b)Normal forces measurements as a function of the applied strain γ0 at a given frequencyf = 1 Hz in parallel plate geometry, (gap is fixed at 2 mm) with rough surfaces.

To further investigate this, we looked in more detail at the rheological behaviour ofthe system. The deviation from Newtonian behaviour of Stokesian suspensions hasmeasured by study of normal stress and the phenomenon of shear-induced particlemigration in suspensions is also closely related to normal stress. From a fundamen-tal perspective, normal stresses in a non-colloidal Stokesian suspension are worthyof study because they are the most important non-Newtonian characteristic it ex-hibits.

To investigate possible normal stresses in the sample and because it is extremely dif-ficult to measure normal forces in Couette geometry, oscillatory measurements weretaken in a circular 50 mm diameter plate-plate geometry. The rheology measuredin the Couette cell is very similar to that in the plate-plate geometry. However,the geometry of the measurement cell will influence both the migration and thestrain at which the normal forces will occur. For the former, the gap size of theCouette cell determines the stress gradient and thus the migration speed [101]. Forthe latter, the confinement is known to influence the transition between the tworegimes [108].

The result for the normal force as a function of the applied strain is shown inFig. 4.5(b): at low strains there is no normal force, but beyond a rather well-defined critical strain, all of a sudden a large normal force emerges. This figurealso shows that that migration is extremely rapid by increasing the volume fraction.The order of magnitude of the normal stress becomes exactly that of the viscousstress; N = FN/A ≈ 0.05/(2.5 × 10−2)2 Pa ≈ 100 Pa (A is the surface area of


the plate) whereas the viscous stress is also around 100 Pa. The fact that thetwo stresses become very similar clearly indicates that beyond the critical strain,we are dealing with a frictional system. However, since our measurements areinherently non-stationary, it is difficult to say something quantitative about thefriction coefficient.

The emergence of the normal stresses is therefore the hallmark of frictional be-haviour; indeed theory predicts a crossover from lubricated (viscous) behaviour tocollisional (frictional) behaviour upon increasing the strain rate [97]. This is alsowhat happens in our experiment: at fixed frequency, an increase in the strain am-plitude also implies an increase in the strain rate. We therefore observe a clear andwell-defined transition, at a critical strain γ0c that is, in addition, similar to thatin the earlier experiments of [89, 90]. It appears evident that if the inter actionsbetween the grains are frictional, their motion can no longer be reversible, as isalso confirmed by simulations on systems similar to the one studied here [98].

0.30 0.35 0.40 0.45 0.50 0.55 0.60

0

50

100

150

200

Crit

ical

stra

in (%

)

Figure 4.6: Critical strain beyond which the normal forces appear as a function of thevolume fraction for the experimental measurements.

The results in this Ref. show that a manybody suspension of near hard spheresin Stokes flow loses reversibility as a result of particle interactions. The observedtendency toward hydroclusters and perhaps solid contact leading to efficient stresstransmission is thus seen to be a natural consequence of the highly anisotropicstructure which causes the particle phase to generate normal stresses.

We studied the dependence of this critical strain γ0c on the volume fraction (Fig.4.6). The data are in reasonable agreement with the earlier findings of Pine et al.using numerical simulations that accounted for the experimental finding of [90]andshows the descending behavior of critical strain with increasing volume fraction.However, in our case, γ0c appears to diverge around a volume fraction of 20%,whereas the data of Pine et al. [90] γ0c diverge for a somewhat lower volume fraction.

4.5. Conclusion 71

We propose the following explanation for the dependence of the critical deformationon the volume fraction. By considering the existence of a frictional regime, wheredynamic contacts are formed, the confinement pressure should play an importantrole [108, 109]. In our system, the grains are confined, both between the platesand in the solvent. The latter provides a confining pressure that is mainly due tothe surface tension of the solvent, making it impossible to remove grains from thesuspension. As suggested by Cates et al. [110], the confinement pressure associatedwith this should be on the order of the surface tension over the grain size. We thenexpect that the critical strain γ0c would correspond to the balance between viscousstress and confinement pressure [103, 107–109] and consequently verify Pconf =η(φ)γ0c = η(φ)γ0cω.

Here, the confinement pressure is due to the fact that the beads are trapped inthe liquid, leading to Pconf ∼ γ/R, with γ ' 20 mN/m the surface tension of theliquid and R the radius of the particles. This directly leads to an estimate for thecritical strain γ0c ∼ 200/(1.4π× η(φ)) as a function of the volume fraction, that isin good agreement with our experimental data (Fig. 4.6), without any adjustableparameters.

4.5 Conclusion

We have shown by a combination of MRI experiments and classical rheology whatthe characteristics are of the irreversible behavior in an oscillatory flow of granularsuspension.

For a low Re non-brownian suspension, however, reversing the boundary motionis equivalent to reversing time. Thus, we have a opportunity to test the limits ofreversibility experimentally: we could undertake a quantitative version of Taylor’sexperiment and measure the degree to which particles return to their initial po-sitions. Here we reported a study of reversibility, which is made by straining aviscous suspension periodically in circular Couette flow and plate-plate geometries.

Even if the total deformation undergone by the samples is zero, during the repeateddeformations particles migrate from the rotating inner cylinder to the stationaryouter cylinder. The migration depends on the accumulated deformation and thuscan not account for the observation of a relatively well-defined critical strain forirreversibility in earlier experiments by Pine et al. [90]. So we present experimentalmeasurements of the normal stresses in sheared suspensions. This was achievedby applying a sinusoidally strain with a fixed frequency, and using a parallel plategeometry.

The experiments show that the interactions in the system can lead to either re-versible or irreversible motion, depending on the amount of deformation that isimposed on the suspension. We found that there is a strain dependent thresholdwhich particles do not return to their starting configurations after some cycles.What can account for the existence of such a critical strain is the onset of frictionalbehavior, as evidenced by the sudden increase of normal stresses at a critical strain


that strongly depends on volume fraction. An important conclusion of the workwas that migration is extremely rapid as φ → φmax. The emergence of normalforces strongly suggests that dynamical contacts are formed in the suspension.

In the absence of nonhydrodynamic particle interactions, the shear stress is linearin the shear rate, yielding a Newtonian shear viscosity. Normal stress can ariseonly if there is anisotropy in the microstructure and irreversibility occurs onlyabove a well-defined threshold strain amplitude, which depends strongly on theconcentration of the suspension. This should be the signature of the formation ofdynamical clusters.

5.

Rheology of suspensions

Abstract: The rheological characterization of suspensions and in general, complexfluids could be challenging because of the sensitive measurement conditions andprocedures. Despite these complexities, because of its obvious importance in awide range of industrial applications, rheology of suspensions has been an importantresearch topic for many years.

This chapter aims to study the behaviour of dense suspensions of non-Brownianparticles in confined geometry. We study the different regimes of suspension flowsand the forces between the grains in a suspension in these regimes. We find thatthe local rheology presents a transition at low shear rate from a viscous to a shear-thickening behaviour with shear stresses proportional to the shear rate squared, aspredicted by a scaling analysis.

5.1 Introduction to the rheology of complex fluids

Complex fluids are immensely important in our life, for industry, understandingcertain biological processes, and so on. Such complex fluids are mostly granu-lar materials, suspensions of particles such as colloids, polymers, or proteins in asolvent. The majority of these suspensions exhibit shear thinning: the faster thematerial flows, the smaller its resistance to flow, or apparent viscosity. If the systemis sheared, the shear pulls the system over certain energy barriers that the systemwould not be able to cross without the applied shear; the viscosity consequentlybecomes small. Because of the generality of the shear-thinning phenomenon, itis interesting to note that exceptions to the rule exist. For certain concentratedsuspensions of particles, shear thickening may be observed as an abrupt increasein the viscosity of the suspension at a certain shear rate [108].

The detailed mechanism of this shear-thickening phenomenon is still under debate[32, 108, 111]. For colloidal suspensions, the phenomenon is often attributed tothe shear-induced formation of hydrodynamic clusters: in this case, the viscosity

73

74 Chapter 5. Rheology of suspensions

increases continuously as a consequence of its dependence on particle configuration;this may be but is not necessarily accompanied by an orderdisorder transition inthe particle configuration.

For granular matter, studies of dry granular materials have evidenced considerableactivity over the last decade. These works were motivated by the importance ofdry granular materials in many practical and industrial situations [97]. So thenumber of studies on wet granular matter is almost negligible compared to that fordry. In the real world, however, we often see wet granular materials, such as beachsand and the cohesion induced by the liquid changes the mechanical properties ofgranular materials. The biggest effect that the liquid in granular media induces isthe cohesion between grains and it is well known that this cohesion depends on theamount of liquid in the system. We discussed different regimes of liquid content inwet granular media in Chapter 1. Suspensions consisting rigid particles in a liquidand are in the category of the slurry regime of wet granular materials.

The rheological behaviour of concentrated suspensions and granular pastes hasbeen dealt with mainly in three fields which consider different types of materialsunder various conditions (which might overlap in some cases): rheology of sus-pensions, physics of granular matter, and soil mechanics. The first field (rheologyof suspensions) was developed by physicists somehow with extrapolating the ap-proach of Einstein concerning dilute suspensions of hard spheres to concentratedsystems [112]. Thus the viscosity of such systems is related to the viscosity of theinterstitial fluid, η0, the solid volume fraction φ, and the maximum packing frac-tion, φm, through various models [113, 114] such as the Krieger-Dougherty modelor Zarraga model [111,115].

Suspension rheology has seen considerable progressdue to both experimental andnumerical theoretical works which have helpedin clarifying the relation betweenmacroscopic stress and the microstructure. However, various disturbing effectsduring the measurements, such as wall slip, sedimentation, migration, and evapo-ration make concentrated suspensions and granular pastes precisely the most dif-ficult systems to study with rheometers. The flow of granular materials has beenan important topic of hot debate and viscosity, as the determinant parameter, isnecessary for many applications to predict the resistance to flow [116].

5.2 Different regimes of sheared granular materi-als

An important topic in studies on dry granular media is the sustained (non-collisional)contacts in flow regimes. So, there is an identification which is correct in extremecases, nearly jammed versus very dilute granular media, that led to the implicitreduction of granular rheology to two limiting situations: quasi-static that is dom-inated by contact forces, and fast flows in which grains interact only during binarycollisions. This issue came into focus after experimental and numerical works pro-vided evidence that Bagnolds scaling (1954) held in various types of dense flows.

5.2. Different regimes of sheared granular materials 75

Suspension rheology has also been considerable progresses, due to both experi-mental and numerical/theoretical works, which have helped clarifying the relationbetween macroscopic stress and the microstructure (Brady and Morris 1997). Thenon-zero values of the normal stress differences observed in both experiments andthe numerical studies are a proof that particle-particle forces are involved [117,118].

5.2.1 Particle-particle contacts in suspensions

Suspension flows can be classified in these three regimes: quasi-static, viscous andinertia. Reynolds (1885) showed that a saturated non-Brownian suspension ina random close packed (RCP) configuration must dilate under shear to sustaincontinued deformation. This phenomenon is called Reynolds dilatation.

At extremely low shear rates and under the application of a normal force, thesystem would be in the quasi-static regime is dominated by short range frictionalinteractions between the particles resulting from extended contact. As the shearrate is increased, the interstitial fluid plays an important role and at sufficientlyhigh shear rates, fluid viscosity will govern the behaviour of the mixture. Thisviscous regime was first studied systematically by Bagnold in 1954, but later stud-ies demonstrated the inaccuracy of Bagnold’s theory. Several theoretical studies(Leighton and Acrivos 1987; Brady and Bossis 1985 [101,119]) have suggested thatnormal stress cannot be generated in Newtonian suspensions in the absence ofparticle contact [120].

Actually Bagnold succeeded to use kinetic theory for the scaling between stressand shear rate (σ ∼ γ2) first observed in dense granular materials [121]. At stillhigher values of shear rate, the flow enters the inertial regime which is dominated bycollisional interactions between the particles and was first discovered by Bagnold in1954. The notion that particle-particle contacts may contribute to stress in a widerange of situations, while their presence is not easily seen via macroscopic scalingrelations, challenges a simplifying view that is consistently found in the literature.In colloidal suspensions, a transition from shear-thinning to shear-thickening canbe seen at low Peclet numbers, Pe = ηγR3/kT , where R is the grains diameterand η is the suspending fluid viscosity [34, 97, 122]. We have discussed this regimein Chapter 1.

For non-Brownian suspensions (typically, R > 10 µm and Pe > 103), the maintypes of macroscopic behaviour such as, shear- thinning, viscous and shear-thickeningare often interpreted as the dominance of one specific form of interactions: contactforces, viscous forces or collisions. Figure 5.1 shows scaling of these forces betweenthe grains. The scale of contact forces is given by:

Fcont ∼ µpR2, (5.1)

where µ is the friction coefficient and p is the pressure between the grains. Whenthe grains come into contact, the roughness of the particles preserves a thin layerof fluid of thickness ε which prevents the divergence of the amplitude of lubricationforces. The scale of viscous forces is thus estimated as:


Figure 5.1: Schematic picture of a suspension and dominant interactions [97].

Fvisc ∼ ηR3γ/ε. (5.2)

Also the collision forces could be estimated as:

Fcolli ∼ mγ2R, (5.3)

where m is the average particle mass.

Determining the transition between these regimes is one of the important debatesin investigating the behaviour of the materials. Dimensionless parameters canprovide criteria to analyse and specify the boundary conditions. To determine theboundaries between these regimes, we can mention two dimensionless parametersthat are constructed from the above estimates: the ratio of viscous force to thecontact force which is called the Leighton number Le = η

µRγεp , and the ratio of

collisional force to the viscous force that is the Bagnold number Ba = mγε/ηR2

[97, 116]. In the frictional regime, no steady flow exists without localization. Sothe scaling and this analysis would suggest that how increasing the shear rate andother parameters affects the dominant forces in the suspension and the transitionbetween different regimes.

5.3 Experimental methods

We studied dense suspensions of non-colloidal spherical particles immersed in aNewtonian fluid. The suspensions were made of spherical polystyrene beads fromDynoseeds with diameter 20 µm, polydispersity < 5%, and density 1050 kg.m−3,suspended in aqueous solutions of NaCl to match the densities of the solvent and

5.4. Results 77

Figure 5.2: (a) parallel plate geometry surrounded by a cylinder that confines thesuspension. (b) Shear stress versus shear rate measured for various gaps and a fixedvolume fraction φ = 56%. The dotted lines are γ scaling and the solid lines are γ2 scaling.

the particle. When the densities of the particles and the fluid are not matched,buoyancy effects appear and density matching ensures that there are no gravityinduced contacts and sedimentations. Also we used a little surfactant to change thewetting of the particles. We varied the volume fraction of the particles, φ = Vg/Vtdefined as the volume of grains Vg on the total volume Vt, from 56% to 58%.

The experiments were carried out with cup-and-plate geometry on a commercialrheometer. The cup-plate geometry is equivalent to plate-plate geometry withlateral surfaces. It enabled us to undertaken the experiments in a confinementsystem with a constant volume of the sample. The diameter of the plate was 50 cmand the two plates had rough surfaces by sandpaper to minimize the wall slip effect(Fig. 5.2 (a)). Another advantage of this geometry is that we can measure thenormal stresses. The rheometer measures a torque T and a rotation rate Ω, whichare related to the stress and shear rate at the edge of the sample by σ = 12T/πD3

and γ = πDΩ/d, where D is the plate diameter and d is the gap between the plates.

5.4 Results

5.4.1 Shear stress behaviour and constitutive equation

A stress-strain curve σ(γ, φ), at a fixed volume fraction is obtained by collecting themeasurements of stress and shear rate for different size of the gaps (Fig. 5.2 (b)).The results show that at a critical shear rate γc, a transition from a Newtonianregime to an inertial regime occurs. Figure 5.3 shows the critical shear rate andcritical shear stress as a function of volume fraction. As we explained about viscousand inertial regimes in section 5.2.1, the scaling for the viscous (Newtonian) regimeshows that σ ∼ γ and in the inertial regime the shear stresse is proportional tothe shear rate squared; σ ∼ γ2. The γ2 scaling identifies a regime where particle


Figure 5.3: (a) Critical shear rate and (b) critical shear stress versus volume fractionfor the viscous-inertial transition.

inertia dominates over viscous forces [97, 103]. Here we investigate the viscous-inertial transition (the mechanism that controls these scaling regimes) and theforces acting on the grains in the wet granular media.

The forces between a set of particles immersed in a viscous interstitial fluid in-troduce hydrodynamic interactions between grains and the contact between rigidgrains. The equation of motion of particles’ center of mass ri, governing a granularsystem is Newton’s equation, which in the absence of external forces is given by:

md2ridt2

=∑j

Fij + F viscij , (5.4)

where m is the mass of particles and Fij and F visci are the rigid contact forcesand hydrodynamic forces, respectively. The rigid forces Fij cannot introduce bydefinition any force or length scale and to perform scaling analysis, we need toseparate the two limiting cases when either inertial or viscous terms dominate.

The first condition specifies a viscous limiting case that occur when viscous forcesare dominant over grain inertia:

∑j Fij + F visci = 0 and inertial regime can be

identified when grain inertia is dominant: md2ri/dt2 =

∑j Fij . Both expressions

imply that in “viscous” regime Fij ∼ γ and in “inertial” regime Fij ∼ γ2 [97, 103,123, 124]. This interpretation shows the existence of a crossover between the twosimple scaling regimes at low shear rate σ ∼ γ (viscous) and high shear rate σ ∼ γ2(inertial). Figure 5.4 shows the shear stress versus shear rate measured for variousvolume fractions. This figure also shows a dependence of critical shear rate onvolume fraction.

Also the scale invariant formalism which lead to an exact scaling analysis of viscousand inertial forces helps to understand why the critical shear rate γc can be so lowat high volume fraction and vanishes precisely at φm [97]. So with the form of

5.4. Results 79

0.1 1 100.01

0.1

1

10

100

Shea

r stre

ss (P

a)

Volume fraction 56% 57% 58%

Shear rate (s-1)

Figure 5.4: Shear stress versus shear rate measured for various volume fractions. Di-ameter of the beads is 20 µm

stresses related to the viscous and inertial regimes, γc vanishes (i) linearly with φ,(ii) at the jamming packing fraction φm.

Figure 5.5 shows the behaviour of critical shear rate of dense suspensions of non-Brownian particles versus volume fraction by macroscopic rheometric experiments[103]. According to this Ref. the crossover between the viscous and inertial regimeswhich is found by equating the two expressions for the stress, finally leading toγc(φ) ∼ (η0/ρR

2)(φm − φ) and σc ∼ η20/ρR2, where ρ and R are the particle

density and diameter, and η0 is the interstitional fluid viscosity. These equationsand the results of Fig. 5.5 show that γc vanishes precisely at the jamming packingfraction φm.

5.4.2 Viscosity of suspension

The notion of viscosity of granular systems and suspensions is necessary for manyapplications to predict the resistance to flow. Figure 5.6 (a) shows the evolutionof viscosity curves for suspensions versus shear rate for different sizes of the gap.At low shear rates, a relatively Newtonian behaviour is observed. We therefore ob-serve a clear and well-defined transition from the Newtonian to the shear-thickeningregime beyond a critical shear rate γc that is in addition similar to previous results.Shear-thickening is a category of non-Newtonian fluid behaviour in which the vis-cosity η defined by η = σ/γ increases as a function of shear rate γ or shear stressσ over some parameter range.

Shear-thickening is generally interpreted as the consequence of dilatancy. Theresults show that for the onset of thickening the smaller gaps leads to the lower


Figure 5.5: critical shear rate for the γ/γ2 transition versus volume fraction [103].

shear rate at which thickening occurs. At a certain shear rate, a very abruptincrease in viscosity is observed, this critical shear rate increases with increasinggap (Fig. 5.6 (b)). Dependency of the critical shear rate for the onset of shear-thickening on the gap of the geometry, can be explained by the tendency of thesheared system to dilate which is a result of collisions between the grains.

It is tempting to see whether the shear-thickening phenomenon itself can be due tothe confinement: if the sample is confined in such a way that the grains cannot rollover each other, this could in principle lead to an abrupt jamming of the system.To check this effect, in the rheometer, instead of setting the gap size for a givenexperiment, one can impose the normal stress and make the gap size vary in orderto reach the desired value of the normal stress and we can see the dependenceof the gap variation on shear rate. Increase in the gap, allowing the system todilate [108,125].

5.5 Normal stresses

The principal information obtaine from the normal stress measurements. The nor-mal stresses are reminiscent of the Reynolds dilatancy of dry granular matter: whensheared, it will dilate in the normal direction of the velocity gradient. Dilatancyis a direct consequence of collisions between the grains: to accommodate the flow,the grains have to roll over each other in the gradient direction, and hence thematerial will tend to dilate in this direction. However, in our system, the grainsare confined, both between the plates and in the solvent.

The latter provides a confining pressure that is mainly due to the surface tensionof the solvent, making it impossible to remove grains from the suspension. Assuggested by Cates et al. [17], the confinement pressure associated with this should

5.6. Conclusion 81

Figure 5.6: (a) Evolution of the viscosity versus shear rate for different size of the gapsat φ = 56%, (b) evolution of the critical shear rate as a function of the gap.

be on the order of the surface tension over the grain size, Pc = γ/R, of the sameorder of magnitude as the typical normal stresses measured in the experiments nearthe onset of shear thickening.

In the parallel plate geometry, the upward force on the rheometer is measured andthe normal stress τN is obtained by dividing this normal force by the plate cross-sectional area. The normal stress τN is shown in Fig. 5.7 as a functions of shearrate γ for the measurements in different volume fractions. At low shear rate thereis no normal force, and the Newtonian regime is dominant on the behaviour of thesuspensions. But beyond a critical shear rate, a sudden large normal force emerges.For this shear-thickening suspensions, we found positive normal stresses, meaningthe sample is pushing against the rheometer plates, in agreement with other mea-surements of shear-thickening [108, 125–127]. When dilation of the granular shearflows is prevented by confinement, shear is instead accompanied by normal forcesagainst the walls.

Hoffman (1982) argued that discontinuous shear- thickening will occur in densesuspensions whenever the particles segregate into layers but are constrained fromrotating as groups below the onset stress. While Wagner showed that in this casesthe layering is not necessary, because the onset is determined by a point where theshear stress is large enough to shear particles in such a way to cause dilation [125].

5.6 Conclusion

We investigated the rheology of dense suspensions of non-Brownian particles inconfined system. When a concentrated colloidal suspension is sheared, particlesorganize. They exhibit an anisotropic microstructure, which in its turn modifiesthe flow properties of the suspension. At low shear rate, this coupling is stable,and leads to permanent ordering of the colloids under shear, for example, along


0.01 0.1 1 10 100

-30

0

30

60

90

120

Shear rate (s-1)

Nor

mal

Stre

ss (P

a)

Volume fraction 56% 57% 58%

Figure 5.7: Normal stress measurements (τN ) vs. shear rate for different volume frac-tions of a sample of 20 µm beads.

planes parallel to the shear direction. When the shear rate becomes high enough,the coupling may become instable. In that case, transient structures develop andlead to a increasing of the suspension viscosity. Indeed, anisotropic microstructureof the particles leads to imbalance of the bulk stress, and thus to nonzero normalstress differences.

Here, we observed a sharp shear-thickening transition at a critical shear rate γcfrom a viscous to a inertial regime with shear stress proportional to the shear ratesquared σ ∼ γ2, as predicted by a scaling analysis. This critical shear rate decreaseswith increacing the volume fraction and vanishes at the jamming packing fractionφm.

Shear-thickening can then be interpreted as the consequence of dilatancy which is adirect result of collisions between the grains. In our system the grains are confined,and to accommodate the flow the grains have to roll over each other and hence thematerial will tend to dilate. For the onset of thickening we observed that the smallgaps of the rheometer, leads to the lower shear rate at which thickening occurs.

We have studied the mechanism of transition between diffrent regimes in suspen-sions and shear-thickening in these sytems, but some details remain unresolvedand this problem warrant further experiments. To investigate possible effects ofthe frictional stress, heterogenity, etc. We intend to carry out more measurementsfor a wide range of experimental conditions.

Bibliography

[1] Abhijit Deshpande, J. Murali Krishnan, and P. B. Sunil Kumar. Rheology ofComplex Fluids. Springer, 2010.

[2] Alexandr Ya. Malkin. Rheology Fundamentals. Chem. Tec. Publishing,Toronto, 1994.

[3] Roger I. Tanner and Kenneth Walters. Rheology: An Historical Perspective,volume 7. Elsevier Science, 1998.

[4] Howard A. Barnes, John F. Hutton, and Kenneth Walters. An Introductionto Rheology, volume 3. Elsevier Science Limited, 1989.

[5] Etienne Guyon. Physical Hydrodynamics. Oxford University Press on De-mand, 2001.

[6] Jan Mewis and Norman J. Wagner. Colloidal Suspension Rheology. Cam-bridge University Press, 2011.

[7] Tasos Papanastasiou, Georgios Georgiou, and Andreas N. Alexandrou. Vis-cous Fluid Flow. CRC Press, 2010.

[8] Philippe Coussot and Norman J. Wagner. The future of suspension rheo-physics: comments on the 2008 workshop. Rheologica Acta, 48(8):827–829,2009.

[9] Kenneth Walters. Rheometry, Industrial Applications, volume 1. ResearchStudies Press, 1980.

[10] Christopher W. Macosko and Ronald G. Larson. Rheology: Principles, Mea-surements, and Applications. VCH New York, 1994.

[11] K. Kesava Rao, Prabhu R. Nott, and S. Sundaresan. An Introduction toGranular Flow. Cambridge University Press Cambridge, 2008.

[12] S. Joseph Antony, William Hoyle, and Yulong Ding. Granular materials:fundamentals and applications. Royal Society of Chemistry, 2004.

[13] Jacques Duran. Sands, Powders, and Grains. Springer New York, 2000.

83

84 BIBLIOGRAPHY

[14] Andrea J. Liu and Sidney R. Nagel. Nonlinear dynamics: Jamming is notjust cool any more. Nature, 396(6706):21–22, 1998.

[15] Daniel T. N. Chen, Qi Wen, Paul A. Janmey, John C. Crocker, and Arjun G.Yodh. Rheology of soft materials. Condensed Matter Physics, 1:301–322,2010.

[16] Reka Albert, Istvan Albert, Daniel Hornbaker, Peter Schiffer, and Albert-Laszlo Barabasi. Maximum angle of stability in wet and dry spherical gran-ular media. Physical Review E, 56(6):R6271–R6274, 1997.

[17] Namiko Mitarai and Franco Nori. Wet granular materials. Advances inPhysics, 55(1-2):1–45, 2006.

[18] Arshad Kudrolli. Granular matter: Sticky sand. Nature Materials, 7(3):174–175, 2008.

[19] Z. Fournier, D. Geromichalos, S. Herminghaus, M. M. Kohonen, F. Mugele,M. Scheel, M. Schulz, B. Schulz, Ch Schier, R. Seemann, and A. Skudelny.Mechanical properties of wet granular materials. Journal of Physics: Con-densed Matter, 17(9):S477, 2005.

[20] Hans-Jurgen Butt. Controlling the flow of suspensions. Science,331(6019):868–869, 2011.

[21] Dennis A. Siginer, Daniel De Kee, and Raj P. Chhabra. Advances in the Flowand Rheology of Non-Newtonian Fluids, volume 8. Elsevier Science, 1999.

[22] Pierre J. Carreau, P. A. Lavoie, and F. Yziquel. Rheological properties ofconcentrated suspensions. Rheology Series, 8:1299–1345, 1999.

[23] K. Kesava Rao, Prabhu R. Nott, and S. Sundaresan. An Introduction toGranular Flow. Cambridge University Press Cambridge, 2008.

[24] Erin Koos and Norbert Willenbacher. Capillary forces in suspension rheology.Science, 331(6019):897–900, 2011.

[25] Christopher D. Willett, Michael J. Adams, Simon A. Johnson, and JonathanP. K. Seville. Capillary bridges between two spherical bodies. Langmuir,16(24):9396–9405, 2000.

[26] Peder C. F. Møller and D. Bonn. The shear modulus of wet granular matter.EPL (Europhysics Letters), 80(3):38002, 2007.

[27] Richard AL Jones. Soft Condensed Matter, volume 6. OUP Oxford, 2002.

[28] Jacob N. Israelachvili. Intermolecular and Surface Forces: revised third edi-tion. Academic Press, 2011.

[29] J. D. Jackson. Classical Electrodynamics. John Wiley & Sons: New York,1998.

BIBLIOGRAPHY 85

[30] David V. Boger. Demonstration of upper and lower newtonian fluid behaviourin a pseudoplastic fluid. Nature, 265(5590):126–128, 1977.

[31] R. P. Chhabra and John Francis Richardson. Non-Newtonian Flow: Funda-mentals and Engineering Applications. Butterworth-Heinemann, 1999.

[32] H. A. Barnes. Shear-thickening (dilatancy) in suspensions of nonaggregatingsolid particles dispersed in newtonian liquids. Journal of Rheology, 33:329,1989.

[33] D. I. Dratler, WR Schowalter, and R. L. Hoffman. Dynamic simulationof shear thickening in concentrated colloidal suspensions. Journal of FluidMechanics, 353:1–30, 1997.

[34] Norman J. Wagner and John F. Brady. Shear thickening in colloidal disper-sions. Physics Today, 62(10):27–32, 2009.

[35] Richard L. Hoffman. Explanations for the cause of shear thickening in con-centrated colloidal suspensions. Journal of Rheology, 42:111, 1998.

[36] Eugene Cook Bingham. Fluidity and Plasticity, volume 1. McGraw-Hill NewYork, 1922.

[37] P. C. F. Møller, A. Fall, and D. Bonn. Origin of apparent viscosity in yieldstress fluids below yielding. EPL (Europhysics Letters), 87(3):38004, 2009.

[38] Daniel Bonn and Morton M. Denn. Yield stress fluids slowly yield to analysis.Science, 324(5933):1401–1402, 2009.

[39] Ning Xu and Corey S. OHern. Measurements of the yield stress in frictionlessgranular systems. Physical Review E, 73(6):061303, 2006.

[40] Q. D. Nguyen and D. V. Boger. Measuring the flow properties of yield stressfluids. Annual Review of Fluid Mechanics, 24(1):47–88, 1992.

[41] C. Tiu, J. Guo, and P. H. T. Uhlherr. Yielding behaviour of viscoplasticmaterials. Journal of Industrial and Engineering Chemistry-Seol, 12(5):653,2006.

[42] J. G. Oldroyd. The elastic and viscous properties of emulsions and suspen-sions. Proceedings of the Royal Society of London. Series A. Mathematicaland Physical Sciences, 218(1132):122–132, 1953.

[43] T. G. Mason and D. A. Weitz. Linear viscoelasticity of colloidal hard spheresuspensions near the glass transition. Physical Review Letters, 75(14):2770–2773, 1995.

[44] Randy H. Ewoldt, Peter Winter, Jason Maxey, and Gareth H. McKinley.Large amplitude oscillatory shear of pseudoplastic and elastoviscoplastic ma-terials. Rheologica Acta, 49(2):191–212, 2010.

86 BIBLIOGRAPHY

[45] Kyu Hyun, Manfred Wilhelm, Christopher O. Klein, Kwang Soo Cho,Jung Gun Nam, Kyung Hyun Ahn, Seung Jong Lee, Randy H. Ewoldt, andGareth H. McKinley. A review of nonlinear oscillatory shear tests: Analy-sis and application of large amplitude oscillatory shear (laos). Progress inPolymer Science, 36(12):1697–1753, 2011.

[46] Seung Ha Kim, Hoon Goo Sim, Kyung Hyun Ahn, and Seung Jong Lee. Largeamplitude oscillatory shear behavior of the network model for associatingpolymeric systems. Korea-Australia Rheology Journal, 14(2):49–55, 2002.

[47] D. Boutelier, Christoph Schrank, and A. Cruden. Power-law viscous materialsfor analogue experiments: New data on the rheology of highly-filled siliconepolymers. Journal of Structural Geology, 30(3):341–353, 2008.

[48] Randy H. Ewoldt, A. E. Hosoi, and Gareth H. McKinley. New measures forcharacterizing nonlinear viscoelasticity in large amplitude oscillatory shear.Journal of Rheology, 52:1427, 2008.

[49] Randy H. Ewoldt and Gareth H. McKinley. On secondary loops in laos viaself-intersection of lissajous-bowditch curves. Rheologica Acta, 49(2):213–219,2010.

[50] Randy H. Ewoldt, Christian Clasen, A. E. Hosoi, and Gareth H. McKinley.Rheological fingerprinting of gastropod pedal mucus and synthetic complexfluids for biomimicking adhesive locomotion. Soft Matter, 3(5):634–643, 2007.

[51] G. D. R. MiDia. On dense granular flows. Eur. Phys. J. E, 14:341–365, 2004.

[52] Pierre Jop, Yoel Forterre, and Olivier Pouliquen. A constitutive law for densegranular flows. Nature, 441(7094):727–730, 2006.

[53] O. Pouliquen, C. Cassar, P. Jop, Y. Forterre, and M. Nicolas. Flow of densegranular material: towards simple constitutive laws. Journal of StatisticalMechanics: Theory and Experiment, 2006(07):P07020, 2006.

[54] Frederic da Cruz, Sacha Emam, Michael Prochnow, Jean-Noel Roux, andFrancois Chevoir. Rheophysics of dense granular materials: discrete simula-tion of plane shear flows. Physical Review E, 72(2):021309, 2005.

[55] Paul R. Day and George G. Holmgren. Microscopic changes in soil structureduring compression. Soil Science Society of America Journal, 16(1):73–77,1952.

[56] Teamrat A. Ghezzehei and Dani Or. Rheological properties of wet soils andclays under steady and oscillatory stresses. Soil Science Society of AmericaJournal, 65(3):624–637, 2001.

[57] Peder C. F. Møller, Jan Mewis, and Daniel Bonn. Yield stress and thixotropy:on the difficulty of measuring yield stresses in practice. Soft Matter, 2(4):274–283, 2006.

BIBLIOGRAPHY 87

[58] Peder C. F. Møller, Abdoulaye Fall, Vijayakumar Chikkadi, Didi Derks, andDaniel Bonn. An attempt to categorize yield stress fluid behaviour. Philo-sophical Transactions of the Royal Society A: Mathematical, Physical andEngineering Sciences, 367(1909):5139–5155, 2009.

[59] Peter Schiffer. Granular physics: a bridge to sandpile stability. NaturePhysics, 1(1):21–22, 2005.

[60] Stephan Herminghaus. Dynamics of wet granular matter. Advances inPhysics, 54(3):221–261, 2005.

[61] Ron Darby. Chemical Engineering Fluid Mechanics. CRC PressI Llc, 2001.

[62] J. E. Fiscina, M. Pakpour, A. Fall, N. Vandewalle, C. Wagner, and D. Bonn.Dissipation in quasistatically sheared wet and dry sand under confinement.Physical Review E, 86(2):020103, 2012.

[63] J. E. Fiscina, Geoffroy Lumay, Francois Ludewig, and Nicolas Vandewalle.Compaction dynamics of wet granular assemblies. Physical Review Letters,105(4):048001, 2010.

[64] M. Scheel, R. Seemann, M. Brinkmann, M. Di Michiel, A. Sheppard, B. Brei-denbach, and S. Herminghaus. Morphological clues to wet granular pile sta-bility. Nature Materials, 7(3):189–193, 2008.

[65] Torsten Groger, Ugur Tuzun, and David M. Heyes. Modelling and measuringof cohesion in wet granular materials. Powder Technology, 133(1):203–215,2003.

[66] W. C. Clark and G. Mason. Tensile strength of wet granular materials. 1967.

[67] Fabien Soulie, Moulay Said El Youssoufi, Fabien Cherblanc, and C. Saix.Capillary cohesion and mechanical strength of polydisperse granular materi-als. The European Physical Journal E, 21(4):349–357, 2006.

[68] Mario Scheel, Dimitrios Geromichalos, and Stephan Herminghaus. Wet gran-ular matter under vertical agitation. Journal of Physics: Condensed Matter,16(38):S4213, 2004.

[69] Peter Sollich, Francois Lequeux, Pascal Hebraud, and Michael E. Cates. Rhe-ology of soft glassy materials. Physical Review Letters, 78(10):2020–2023,1997.

[70] Jorge E. Fiscina and Manuel O. Caceres. Fermi-like behavior of weakly vi-brated granular matter. Physical Review Letters, 95(10):108003, 2005.

[71] Ping Wang, Chaoming Song, and Hernan A. Makse. Dynamic particle track-ing reveals the ageing temperature of a colloidal glass. Nature Physics,2(8):526–531, 2006.

88 BIBLIOGRAPHY

[72] Kevin Lu, E. E. Brodsky, and H. P. Kavehpour. A thermodynamic unificationof jamming. Nature Physics, 4(5):404–407, 2008.

[73] Peter Sollich and Michael E Cates. Thermodynamic interpretation of softglassy rheology models. Physical Review E, 85(3):031127, 2012.

[74] Vincent Richefeu, Moulay Saıd E. l. Youssoufi, and Farhang Radjai.Shear strength properties of wet granular materials. Physical Review E,73(5):051304, 2006.

[75] Maryam Pakpour, Mehdi Habibi, Peder Møller, and Daniel Bonn. How toconstruct the perfect sandcastle. Scientific reports, 2(549), 2012.

[76] Mika M. Kohonen, Dimitrios Geromichalos, Mario Scheel, Christof Schier,and Stephan Herminghaus. On capillary bridges in wet granular materials.Physica A: Statistical Mechanics and its Applications, 339(1):7–15, 2004.

[77] Sarah Nowak, Azadeh Samadani, and Arshad Kudrolli. Maximum angle ofstability of a wet granular pile. Nature physics, 1(1):50–52, 2005.

[78] D. J. Hornbaker, Reka Albert, Istvan Albert, Albert-Lazlo Barabasi, andPeter Schiffer. What keeps sandcastles standing? Nature, 387(6635):765,1997.

[79] P. Tegzes, T. Vicsek, and P. Schiffer. Avalanche dynamics in wet granularmaterials. Physical Review Letters, 89(9):094301, 2002.

[80] Thomas C. Halsey and Alex J. Levine. How sandcastles fall. Physical ReviewLetters, 80(14):3141–3144, 1998.

[81] David Quere. Non-sticking drops. Reports on Progress in Physics,68(11):2495, 2005.

[82] C. M. Wang, C. Y. Wang, and J. N. Reddy. Exact Solutions for Buckling ofStructural Members. CRC Series in Computational Mechanics and AppliedAnalysis. CRC Press LLC, Boca Raton, FL, 2005.

[83] Brian G. Falzon. Buckling and Post Buckling Structures: Experimental, An-alytical and Numerical Studies, volume 1. Imperial College Press, 2008.

[84] www.gadling.com/2007/06/11/the-worlds-tallest-sandcastle-and-other-sandsculpture-marvels.

[85] Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz. Theory of elas-ticity. Course of Theoretical Physics, 7:3, 1986.

[86] Steven J. Cox and C. McCarthy. The shape of the tallest column. SIAMJournal on Mathematical Analysis, 29(3):547–554, 1998.

[87] www.sandworld.de.

BIBLIOGRAPHY 89

[88] Ed J. Vitz and Richard Cornellus. Magic sand: modelling the hydropho-bic effect and reversed-phase liquid chromatography. Journal of ChemilcalEducation, 67(6):512–515, 1990.

[89] Jerry Gollub and David Pine. Microscopic irreversibility and chaos. PhysicsToday, 59(8):8, 2006.

[90] D. J. Pine, J. P. Gollub, J. F. Brady, and A. M. Leshansky. Chaos andthreshold for irreversibility in sheared suspensions. Nature, 438(7070):997–1000, 2005.

[91] Laurent Corte, P. M. Chaikin, J. P. Gollub, and D. J. Pine. Random orga-nization in periodically driven systems. Nature Physics, 4(5):420–424, 2008.

[92] Laurent Corte, S. J. Gerbode, W. Man, and D. J. Pine. Self-organized criti-cality in sheared suspensions. Physical Review Letters, 103(24):248301, 2009.

[93] Gustavo During, Denis Bartolo, and Jorge Kurchan. Irreversibility andself-organization in hydrodynamic echo experiments. Physical Review E,79(3):030101, 2009.

[94] Jeffrey S. Guasto, Andrew S. Ross, and Jerry P. Gollub. Hydrodynamicirreversibility in particle suspensions with nonuniform strain. Physical ReviewE, 81(6):061401, 2010.

[95] Steven Slotterback, Mitch Mailman, Krisztian Ronaszegi, Martin van Hecke,Michelle Girvan, and Wolfgang Losert. Onset of irreversibility in cyclic shearof granular packings. Physical Review E, 85(2):021309, 2012.

[96] Daniel Bonn, Stephane Rodts, Maarten Groenink, Salima Rafai, NoushineShahidzadeh-Bonn, and Philippe Coussot. Some applications of magneticresonance imaging in fluid mechanics: complex flows and complex fluids.Annu. Rev. Fluid Mech., 40:209–233, 2008.

[97] Anael Lemaıtre, Jean-Noel Roux, and Francois Chevoir. What at do drygranular flows tell us about dense non-brownian suspension rheology? Rhe-ologica acta, 48(8):925–942, 2009.

[98] Jeffrey F. Morris. A review of microstructure in concentrated suspensions andits implications for rheology and bulk flow. Rheologica Acta, 48(8):909–923,2009.

[99] Stephen Blundell and David Thouless. Magnetism in Condensed Matter,volume 1. Oxford University Press New York, 2001.

[100] Tal Geva. Magnetic resonance imaging: historical perspective. Journal ofCardiovascular Magnetic Resonance, 8(4):573–580, 2006.

90 BIBLIOGRAPHY

[101] David Leighton and Andreas Acrivos. Shear-induced migration of particlesin concentrated suspensions. Journal of Fluid Mechanics, 181(1):415–439,1987.

[102] Nicolas Huang and Daniel Bonn. Viscosity of a dense suspension in couetteflow. Journal of Fluid Mechanics, 590(1):497–507, 2007.

[103] Abdoulaye Fall, Anael Lemaıtre, Francois Bertrand, Daniel Bonn, and Guil-laume Ovarlez. Shear thickening and migration in granular suspensions. Phys-ical Review Letters, 105(26):268303, 2010.

[104] Daniel Lhuillier. Migration of rigid particles in non-brownian viscous suspen-sions. Physics of Fluids, 21:023302, 2009.

[105] Jeffrey F. Morris and Fabienne Boulay. Curvilinear flows of noncolloidalsuspensions: the role of normal stresses. Journal of Rheology, 43:1213, 1999.

[106] Prabhu R. Nott and John F. Brady. Pressure-driven flow of suspensions:simulation and theory. Journal of Fluid Mechanics, 275(1):157–199, 1994.

[107] Abdoulaye Fall, Francois Bertrand, Guillaume Ovarlez, and Daniel Bonn.Yield stress and shear banding in granular suspensions. Physical ReviewLetters, 103(17):178301, 2009.

[108] Abdoulaye Fall, N. Huang, F. Bertrand, G. Ovarlez, and Daniel Bonn. Shearthickening of cornstarch suspensions as a reentrant jamming transition. Phys-ical Review Letters, 100(1):018301, 2008.

[109] Claire Bonnoit, Jose Lanuza, Anke Lindner, and Eric Clement. Mesoscopiclength scale controls the rheology of dense suspensions. Physical ReviewLetters, 105(10):108302, 2010.

[110] M. E. Cates, M. D. Haw, and C. B. Holmes. Dilatancy, jamming, and thephysics of granulation. Journal of Physics: Condensed Matter, 17(24):S2517,2005.

[111] Ronald G. Larson. The Structure and Rheology of Complex Fluids, volume 2.Oxford University Press New York, 1999.

[112] P. Coussot and Ch Ancey. Rheophysical classification of concentrated sus-pensions and granular pastes. Physical Review E, 59(4):4445, 1999.

[113] Richard Buscall. Effect of long-range repulsive forces on the viscosity ofconcentrated latices: comparison of experimental data with an effective hard-sphere model. J. Chem. Soc., Faraday Trans., 87(9):1365–1370, 1991.

[114] Richard Buscall. An effective hard-sphere model of the non-newtonian viscos-ity of stable colloidal dispersions: comparison with further data for stericallystabilised latices and with data for microgel particles. Colloids and SurfacesA: Physicochemical and Engineering Aspects, 83(1):33–42, 1994.

BIBLIOGRAPHY 91

[115] Isidro E. Zarraga, Davide A. Hill, and David T. Leighton Jr. The character-ization of the total stress of concentrated suspensions of noncolloidal spheresin newtonian fluids. Journal of Rheology, 44:185, 2000.

[116] N. Huang, G. Ovarlez, F. Bertrand, S. Rodts, P. Coussot, and Daniel Bonn.Flow of wet granular materials. Physical Review Letters, 94(2):028301, 2005.

[117] John F. Brady and Jeffrey F. Morris. Microstructure of strongly shearedsuspensions and its impact on rheology and diffusion. Journal of Fluid Me-chanics, 348:103–139, 1997.

[118] A. Sierou and J. F. Brady. Rheology and microstructure in concentratednoncolloidal suspensions. Journal of Rheology, 46:1031, 2002.

[119] John F. Brady and Georges Bossis. Rheology of concentrated suspensionsof spheres in simple shear flow by numerical simulation. Journal of FluidMechanics, 155:105–29, 1985.

[120] D. Prasad and H. K. Kytomaa. Particle stress and viscous compaction dur-ing shear of dense suspensions. International Journal of Multiphase Flow,21(5):775–785, 1995.

[121] Ralph A. Bagnold. Experiments on a gravity-free dispersion of large solidspheres in a newtonian fluid under shear. Proceedings of the Royal Societyof London. Series A. Mathematical and Physical Sciences, 225(1160):49–63,1954.

[122] William B. Russel, Dudley Albert Saville, and William Raymond Schowalter.Colloidal Dispersions. Cambridge University Press, 1992.

[123] Anael Lemaitre. Origin of a repose angle: kinetics of rearrangement forgranular materials. Physical Review Letters, 89(6):064303, 2002.

[124] Gregg Lois, Anael Lemaıtre, and Jean M. Carlson. Numerical tests of consti-tutive laws for dense granular flows. Physical Review E, 72(5):051303, 2005.

[125] Eric Brown and Heinrich M. Jaeger. The role of dilation and confining stressesin shear thickening of dense suspensions. Journal of Rheology, 56:875, 2012.

[126] Didier Lootens, Henri Van Damme, Yacine Hemar, and Pascal Hebraud.Dilatant flow of concentrated suspensions of rough particles. Physical ReviewLetters, 95(26):268302, 2005.

[127] Didier Lootens, Henri Van Damme, Pascal Hebraud, et al. Giant stress fluc-tuations at the jamming transition. Physical Review Letters, 90(17):178301–178301, 2003.

Summary

This thesis is dedicated to the study of the Rheology of Dry, Wet and PartiallySaturated Granular Materials. Granular media, suspensions, emulsions, polymersand gels are ubiquitous in the chemical and materials processing industry, anddespite their very different appearance, the rheology and study of the behaviour ofthese materials is the key to the large-scale industrial production.

Granular materials are large collections of discrete particles. A granular material iscalled dry if the fluid in the interstices or voids between the grains is a gas, whichis usually air. For dry granular media, the dominant interactions are inelasticcollisions and friction, which are short range and non-cohesive. If the voids arecompletely filled with a liquid, the material is called a saturated granular material.If there is a liquid in some of the voids, and the rest of the voids are filled witha gas, the material is said to be partially saturated. Surprisingly, adding a smallamount of liquid to granular matter transforms its properties because the liquidinduces a cohesion between the grains. Cohesion in wet granular media arises fromthe surface tension and capillary effects of the liquid. The mechanical propertiesat low water contents are determined by the liquid bridges between grains, andthose at high water content are determined by the flow of the liquid through thesoil pores.

We aim here to study the behaviour of granular materials in different regimesof wetness. We have investigated the flow, stability, optimum strength, time re-versibility and other rheological behaviour of these systems (dry, partially granularmaterials and suspensions) in different experiments.

In the first part of the thesis we study the behaviour of “dry and partially saturatedgranular materials”. To investigate their mechanical properties, we compared therheology of wet and dry granular materials near the jamming transition for differentsizes of the grains and different volume fractions. Our samples were composed ofglass spheres of diameter d = 140, 250 and 500 µm with and without small amountsof liquid. Partially saturated sand has a much higher yield stress and shouldtherefore have a much higher apparent viscosity for slow flows. For this reason, itis commonly believed that wet sand should show a larger resistance to flow, i.e.,more viscous, than dry sand. We found, however, that in two very different setups(a shear cell which quasi-statically pushed the sand and a cup-plate rotationalrheometer applying large amplitude oscillatory shear), the energy dissipation ofdry sand is larger than that of wet sand. So it is much easier to push wet sandthan dry granular matter. We showed that this is due to the fact that the adhesionbetween the grains decreases the confining pressure and hence decreases the flowresistance. Even if the capillary forces increase the yield stress, the water promotescluster formation and reduces effective intergrain friction, whereas for dry sandthe yield stress is zero in the absence of gravity and a pure frictional behaviour isobserved.

92

93

In the next chapter, to obtain a deeper insight into the effects of liquid content onthe stability of granular materials, we studied the stability of wet sand columnsand the optimum strength of these systems. This allows, amongst other things, topredict the maximum height of a sandcastle. A column becomes unstable to elasticbuckling under its own weight. To verify this experimentally, beach sand with anaverage radius of 100 µm was mixed with a small amount of deionized water. Wefound that the maximum height of the sand column, increases to the 2/3 power ofthe base radius of the column. Measuring the elastic modulus of the wet sand, wefound the optimum strength of sand versus the liquid volume fraction.

In the second part of the thesis, we investigate the behaviour of completely “wetgranular materials”, non-Brownian suspensions with different volume fractions ofparticles were studied in our measurements.

We find that the particle motion becomes irreversible when the particles are sub-jected to a large-amplitude oscillatory shear, when the deformation exceeds a crit-ical value. The origin of the irreversibility is still debated. By a combination ofMRI experiments and classical rheology, we uncover the origin of the irreversiblebehaviour in an oscillatory flow of granular suspension. These methods can probethe homogeneity of the suspensions and reveals an irreversible migration of par-ticles from high-into low shear rate regions. Also we found that above a criticaldeformation, large normal forces appear in the suspensions. The onset of frictionalbehaviour can then account for the existence of such a critical strain as evidencedby the sudden increase in normal stresses that strongly depend on the volume frac-tion. Such contacts also lead to irreversibility in the motion of the particles, andin addition give a quantitative criterion for the onset of irreversibility that agreeswith the experiments.

In the last chapter we study the rheology of dense suspensions of non-Brownianparticles in a confined geometry under imposed gap. Dense suspensions exhibit abehavior known as shear thickening in which the viscosity jumps up dramatically.Measurements were performed with a rotational rheometer and a cup-plate geome-try. The suspensions are made of spherical polystyrene beads with diameter 20 µmsuspended in aqueous solutions. Our results show a transition at low shear ratefrom viscous to a shear thickening behaviuor with shear stresses proportional tothe shear rate squared, as predicted by a scaling analysis.

94

Samenvatting

Samenvatting Dit proefschrift is gewijd aan de studie van De reologie van droge,natte en deels verzadigde granulaire materialen. Granulaire media, suspensies,emulsies, polymeren en gels zijn veel voorkomende materialen in onder anderede chemische en materiaal verwerkingsindustrie. Het bestuderen van hun gedrag,waaronder hun reologische eigenschappen, is essentieel voor hun grootschalige in-dustrile verwerking en productie.

Granulaire materialen zijn grote agglomeraties van discrete deeltjes. Een granulairmateriaal wordt droog genoemd als de ruimtes tussen de korrels gevuld is met eengas, meestal lucht. Bij droge granulaire materialen zijn inelastische botsingen enwrijving tussen de deeltjes de voornaamste interacties. Deze benvloeden de deelt-jes als deze dicht bij elkaar komen maar zorgen er niet voor dat ze blijven plakkenals ze elkaar raken. Als de ruimtes tussen de deeltjes volledige gevuld zijn meteen vloeistof dan worden granulaire materialen verzadigd genoemd. Is maar eendeel van de ruimtes gevuld met een vloeistof en de rest met een gas dan heet hetmateriaal deels verzadigd. Verassend genoeg zorgt het toevoegen van een kleinehoeveelheid vloeistof er al voor dat de eigenschappen van het granulaire materi-aal compleet veranderen. Dit komt doordat er een aantrekkingskracht ontstaattussen de deeltjes. Deze aantrekkingskracht wordt in natte granulaire materialenveroorzaakt door de oppervlakte spanning van de vloeistof. Als er weinig waterin het materiaal zit dan worden de mechanische eigenschappen bepaald door devloeistofbruggen tussen de korrels, terwijl bij veel water deze juist bepaald wordendoor de stroming van het water door de porin tussen de deeltjes.

Ons doel is om het gedrag van granulaire materialen met verschillende hoeveel-heden water te bestuderen. We hebben de stroming, stabiliteit, optimale sterkte,omkeerbaarheid in de tijd en andere reologische eigenschappen van deze systemen(droog, nat en deels verzadigde granulaire systemen en suspensies) in verschillendeexperimenten onderzocht.

In het eerste deel van dit proefschrift hebben we het gedrag van droge en deelsverzadigde granulaire materialen onderzocht. Om hun mechanische eigenschap tebepalen werd de mechanische eigenschappen van natte en droge granulaire ma-terialen met korrels van verschillende grote en bij verschillende volume fractiesonderzocht vlak bij het punt waarop de deeltjes net vast komen te zitten (jammingovergang). Onze monster bestonden uit glazen bolletjes met een diameter vand = 140, 250 en 500 µm, met en zonder een kleine hoeveelheid vloeistof daaraantoegevoegd. Een deels verzadigd granulair materiaal heeft een veel hogere yieldstress (afschuifspanning die op het materiaal aangebracht moet worden voordathet gaat stromen) en zou daardoor een veel hogere viscositeit moeten hebben bijlage stroomsnelheden. Door gebruik te maken van twee geheel verschillende op-stellingen (een shear-cell welke het zand quasi-statisch duwt en een “cup-plate”rotatie reometer welke een oscillerende shear deformatie induceert) hebben wij

95

aangetoond dat de dissipatie van energie in droog zand veel hoger is dan in natzand. Ook zagen we dat het veel makkelijker is om nat zand door een buis teduwen in een Poiseuille-achtig profiel dan droog zand. Dit komt omdat de adhesietussen de korrels de normaalspanning in het stomende zand verlaagd en daardoorook de weerstand om te stromen. Zelfs als de capillaire krachten tussen de deeltjesde “yield-stress” verhogen, dan nog zorgt het water er voor dat er meer clustersgevormd worden. Dit verlaagt de effectieve wrijving tussen de korrels. Voor droogzand vinden we echter geen “yield-stress” en het gedrag wordt alleen bepaald doorde wrijving.

In het volgende hoofdstuk bestudeerden we de stabiliteit van kolommen van natzand en de optimale sterkte van deze systemen om een beter inzicht te krijgenin het effect van toegevoegde vloeistof op de mechanische sterkte van granulairematerialen. We onderzoeken onder andere wanneer een kolom zand instabiel wordtonder zijn eigen gewicht. Om dit experimenteel te verifiren werd strand zand meteen gemiddelde radius van 100 µm gemengd met kleine hoeveelheden gede-ionizeerdwater. We vonden dat de maximale hoogte van de zandkolom met een macht van2/3 toeneemt ten opzichte van de straal van de basis van de kolom. Door deelastische modulus van het natte zand te meten vonden we een relatie tussen deoptimale sterkte van het zand en de volumefractie vloeistof. In het tweede deel vanhet proefschrift onderzochten we het gedrag van natte granulaire materialen. Inonze metingen werden suspensies bestudeerd met verschillende volume fracties enwaarvan de deeltjes geen Brownse beweging vertoonden.

De beweging van de deeltjes wordt irreversibel wanneer de deeltjes worden bloot-gesteld aan een oscillerende shear, wanneer de deformatie groter wordt dan eenkritische waarde. Door gebruik te maken van een combinatie van MRI exper-imenten en klassieke reologie, onderzoeken we de karakteristieken van het irre-versibele gedrag van granulaire suspensies in een oscillerende stroming. We vindendat er een irreversibele migratie van deeltjes van lage naar hoge shear gebieden latenzien. We vinden ook dat boven een zekere kritische deformatie er grote normaalkrachten zichtbaar worden in de suspensies. Het bestaan van een kritieke defor-matie zoals zichtbaar in de plotselinge verhoging in de normaal krachten (welkesterk afhangen van de volumefractie) kan verklaard worden door wrijvingsgedrag.Dit soort contacten tussen de deeltjes resulteren ook in de onomkeerbaarheid vande beweging van de deeltjes en geven bovendien een kwantitatief criterium voor deaanvang van irreversibiliteit in overeenstemming met de experimenten.

In het laatste hoofdstuk bestuderen we de reologie van geconcentreerde suspensiesvan niet-Brownse deeltjes in een ingesloten geometrie onder een zogenaamde “im-posed gap”. Suspensies met een hoge dichtheid vertonen gedrag dat bekend staatals “shear-thickening” waarbij de viscositeit dramatische omhoog schiet wanneer desnelheidsgradient opgevoerd wordt. Metingen werden uitgevoerd met een rotatiereometer met een “cup-plate” geometrie. De suspensies bestonden uit polystyreenbolletjes met een diameter van 20 µm in water. Onze resultaten laten een overgangzien van visceus gedrag bij lage “shear” naar “shear-thinning” gedrag bij hogere“shear” waarbij de shear-stress proportioneel is aan de “shear rate” tot de macht2, zoals ook wordt voorspeld door schalings-analyse.

96

Nomenclature

Symbol Physical Quantity

τN Normal stress [Pa]

η Viscosity [Pa.s]

γ Strain [-]

γ Shear rate [s−1]

T Torque [N.m]

N1, N2 Normal stress differences [Pa]

φ Volume fraction [-]

ρ Density [kg.m−3]

A Hamaker constant [J ]

kB Boltzmann constant [J.K−1]

D Diffusion coefficient [m2s−1]

Pe Peclet number [-]

Re Reynolds number [-]

τ Relaxation time [s]

G′ Shear storage modulus [Pa]

G′′ Shear loss modulus [Pa]

E Young’s modulus [Pa]

Ω Angular velocity [rad.s−1]

Ed Energy dissipated [J ]

M Magnetization [Am−1]

P Pressure [Pa]

µ Friction coefficient [-]

g Gravitational acceleration [m.s−2]

97

List of Publications

1. How to construct the perfect sandcastle, M. Pakpour, M. Habibi,P. Møller, D. Bonn, Sci. Rep. 2, 549 (2012).

2. Dissipation in quasistatically sheared wet and dry sand under con-finement, J. E. Fiscina, M. Pakpour, A. Fall, N. Vandewalle, C.Wagner, D. Bonn, Phys. Rev. E. 86, 020103(R) (2012).

3. Sliding friction on sand, A. Fall, B. Weber, M. Pakpour, N. Shahidzadeh,J. Fiscina, C. Wagner, D. Bonn, submitted.

4. Flow irreversibility in granular suspensions under large amplitudeoscillatory shear, C. Bonnoit, M. Pakpour, F. Bertrand, D. Bonn,in preparation.

5. Drop formation of particles suspension, M. Aytouna, B. Vos, M.Pakpour, D. Bonn, in preparation.

6. Spreading of a non-viscous drop on a viscous substrate, M. Pakpour,M. Habibi, D. Bonn, in preparation.

98

Acknowledgments

I am greatly indebted to Prof. Y. Sobouti and Prof. M. R. Khajehpour, thefounders of the Institute for Advanced Studies in Basic Sciences (IASBS) and Dr.H. R. Khalesifard, the former Vice-Chancellor of the IASBS for giving me thisopportunity to do my PhD thesis under the supervision of Prof. Daniel Bonn.

I would like to thank my supervisor Prof. Daniel Bonn. I have learned an immenseamount from his perspective on research, his sharp insight on almost any issues,and his high expectations of excellence. I would also like to thank Dr. MehdiHabibi for all his help.

It was a great pleasure to work with Dr. Abdoulaye Fall. I particularly enjoyeduseful discussions and also nice experiences to work with him in the Rheologylaboratory of the Van der Waals-Zeeman institute for one month. I am very gratefulto Dr. Jorge Fiscina for all the discussions we had. It was great pleasure to workwith all the member of the soft matter group in the WZI. I would like to thankthe committee members for their positive comments enabling me to improve thethesis.

I would like to express my warm thanks to Rojman who was the first person I metat the WZI and helped me a lot to get used to my new life in Amsterdam. I amalso very grateful to Sareh and all my kind friends at the WZI and the IASBS. Ihad such a great time and a stimulating environment with them during my PhD.I like to acknowledge Sandra Veen who helped me for the Dutch summary of mythesis.

And finally my sincere gratitude to Alireza who was always by my side through allthe happiness and hardship during my PhD period.

uva-dare (digital academic repository) rheology of dry, partially … · the de nition of rheology...

Documents