Stability, Avalanches, and Flow in Dry and Wet Granular ... - CiteSeerX

DISSERTATION for the degree of DOCTOR OF PHILOSOPHY in PHYSICS

Stability, Avalanches, and Flow in Dry and Wet Granular Materials ´ T EGZES P AL

Graduate School of Physics Head: Prof. Zalán Horváth Statistical Physics, Biological Physics, and Physics of Quantum Systems Program Head: Prof. Tamás Vicsek Advisor: Prof. Tamás Vicsek

Department of Biological Physics Eötvös University Budapest, Hungary 2002

Contents Introduction

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PART I :

OVERVIEW

1 A Physicist’s Viewpoint of Granular Materials 1.1 Playing in the Sand – Granular Experiments 1.1.1 Resting State . . . . . . . . . . . . 1.1.2 Sound Propagation . . . . . . . . . 1.1.3 Vibrational Excitations . . . . . . . 1.1.4 Segregation Effects . . . . . . . . . 1.1.5 Deformations and Flows . . . . . . 1.1.6 Clustering in Granular Gases . . . . 1.2 Difficulties of Classical Approaches . . . . 1.2.1 Statistical Mechanics . . . . . . . . 1.2.2 Hydrodynamics . . . . . . . . . . . 1.2.3 Classical Continuum Description . 1.3 Directions of Granular Physics . . . . . . . 1.3.1 Phenomenological Models . . . . . 1.3.2 Characterization of Materials . . . . 1.3.3 Continuum theories . . . . . . . . . 1.3.4 Thermodynamical principles . . . . 1.3.5 Granular flow models . . . . . . . 1.3.6 Microscopical theories . . . . . . . 1.3.7 Computer Simulations . . . . . . . 1.4 Novel Concepts and Fruitful Analogies . . . 1.4.1 Self-Organized Criticality . . . . . 1.4.2 Slow Relaxations . . . . . . . . . . 1.4.3 Chaos . . . . . . . . . . . . . . . . 1.4.4 Traffic Models . . . . . . . . . . . 1.4.5 Granular Ratchets . . . . . . . . . .

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C ONTENTS 2 Continuous and Stick-Slip Motion in Granular Media 2.1 Stick-Slip Processes in Nature . . . . . . . . . . . . 2.1.1 Definition and Examples . . . . . . . . . . . 2.1.2 Minimal Model . . . . . . . . . . . . . . . . 2.1.3 Transition to Continuous Motion . . . . . . . 2.1.4 Stick-Slip Processes in Granular Materials . . 2.2 The Sticking Phase — Stability of Jammed States . . 2.2.1 Jamming . . . . . . . . . . . . . . . . . . . 2.2.2 Stability of Dry and Wet Granular Materials . 2.2.3 Aging . . . . . . . . . . . . . . . . . . . . . 2.3 The Slip Process — Internal and Surface Avalanches 2.3.1 Avalanche Size Distribution . . . . . . . . . 2.3.2 Avalanche Dynamics . . . . . . . . . . . . . 2.3.3 Dilation . . . . . . . . . . . . . . . . . . . . 2.4 Continuous Motion — Granular Flows . . . . . . . .

PART II :

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NEW RESULTS

3 Stick-Slip Fluctuations in Granular Drag 3.1 Introduction . . . . . . . . . . . . . . 3.2 The Experimental Setup . . . . . . . 3.3 Stick-slip Internal Friction . . . . . . 3.4 Transition to “Stepped” Fluctuations . 3.5 Discussion . . . . . . . . . . . . . .

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4 Liquid-Induced Transitions in Granular Media 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 The Experimental Setup . . . . . . . . . . . 4.3 Three Regimes of Behaviour . . . . . . . . . 4.3.1 The Repose Angle . . . . . . . . . . 4.3.2 The Draining Process . . . . . . . . . 4.3.3 Fluctuations . . . . . . . . . . . . . . 4.4 Comparison to Theories . . . . . . . . . . . .

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5 Avalanche Dynamics in Wet Granular Materials 5.1 The Rotating Drum Experiment . . . . . . . . . . . . . . . 5.2 Measurements of the surface angle . . . . . . . . . . . . . . 5.2.1 Flow Phases . . . . . . . . . . . . . . . . . . . . . 5.2.2 Variation of the Surface Angle during an Avalanche . 5.3 Stick-slip Model for the Avalanches . . . . . . . . . . . . . 5.3.1 Definition of the Model . . . . . . . . . . . . . . . . 5.3.2 Comparison to Experiments . . . . . . . . . . . . . II

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C ONTENTS 5.4

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The Phase Diagram . . . . . . . . . . . . . . . . . 5.4.1 Surface Shape . . . . . . . . . . . . . . . . 5.4.2 Aging . . . . . . . . . . . . . . . . . . . 5.4.3 Effect of Parameters on the Phase Diagram Avalanche Dynamics . . . . . . . . . . . . . . . . 5.5.1 Avalanche types . . . . . . . . . . . . . . 5.5.2 Pattern formation . . . . . . . . . . . . . 5.5.3 Larger Beads . . . . . . . . . . . . . . . . 5.5.4 Statistics of Avalanche Sizes . . . . . . . . Continuous Flows . . . . . . . . . . . . . . . . . . 5.6.1 Velocity at the Surface . . . . . . . . . . . 5.6.2 Flow Depth . . . . . . . . . . . . . . . . . 5.6.3 Velocity Profile . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . .

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77 79 81 82 83 84 89 89 91 92 92 94 95 100

Acknowledgement

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A Laboratory Exercise for Physicist Students

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B Image Processing Techniques B.1 Determining the Surface Profile . . . B.2 Determining the Velocity Fields . . . B.2.1 Particle Imaging Velocimetry B.2.2 Following Tracer Particles . .

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References

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Summary

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¨ Osszefoglal´ o

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III

Introduction The sand crunching under our feet, the sugar sprinkled on our donut, or the detergent poured into the washing machine; granular materials are present everywhere in our daily life. They are used, eaten, poured, packed, shaken, and mixed every day. Even little children know quite a lot about granular materials, and we may remember many observations from our childhood experience. If we shake muesli we can get raisins or big flakes to the surface. A vacuum-sealed pack of coffee is quite rigid, but when the pack has been opened, it can be deformed easily. We can draw figures into sand by a stick, and then we can wipe them out with a single movement of the hand. If water is added to the sand or if it is compactified it by tapping, much steeper sandcastles can be built. We can walk on sand and a rod stuck into it can hold a tent, but if we take it to our hands then it flows out among our fingers. These phenomena are simple, but still, physicists — equipped with all the tools of modern physics — often have a hard time trying to give deep and quantitative explanations to them. Although there have been successful attempts to describe the behaviour of granular materials in specific situations, the construction of a generally applicable model is apparently beyond our knowledge. Granular materials behave sometimes like a solid, sometimes like a liquid, but most often differently from both, they are sometimes simple other times complex, sometimes predictable other times surprising; their behaviour is extremely diverse and rich. The ongoing enthusiastic experimental research has been discovering a great variety of interesting phenomena, always offering new challenges to theoretical efforts. One of the most characteristic properties of granular systems is that they tend to freeze into metastable states, often referred to as jammed states, where the motion of every individual grain is constrained by its neighbours. This property enables us to walk and build houses on sand, and this allows sandpiles and dunes to maintain a non-horizontal surface. In case of slow deformations the system often goes through a succession of jammed states with quick rearrangements between them. This type of behaviour belongs 1

I NTRODUCTION to the so called stick-slip processes: the “sticking” phase consists of a jammed state, and during the “slip” this jammed state fails and a new one is formed. In present work I report on experiments investigating the formation, stability, and collapse of jammed states in granular materials. First a general introduction to granular materials is given (Chapter 1). Then I summarize concisely the earlier studies of stick-slip type and continuous motion in granular materials and their relation to granular avalanches and surface flows (Chapter 2). The rest of the thesis deals with our own experiments. Chapter 3 investigates the dynamic evolution of jamming in granular media through fluctuations in the granular drag force, i.e. the force resisting a solid object being pulled slowly through a granular medium. Then I investigate granular samples with inclined surfaces, which are spectacular examples of jammed states, and the dynamics of avalanches, which — in this context — are impressive slip events. In particular I address the question of how the addition of a small amount of liquid modifies the statics and dynamics of these systems. In Chapter 4 I analyse the increased stability of granular samples in the presence of an interstitial liquid due to the formation of liquid bridges among the grains. The final chapter (Chapter 5) presents a detailed study of the dynamics of avalanching wet granular media in a rotating drum apparatus.

2

Part I OVERVIEW

3

Chapter 1 A Physicist’s Viewpoint of Granular Materials Materials in a granular state are of great technological importance which has made them a subject of intense engineering research for a long time. In recent years these materials have attracted a considerable interest also in the physicist community. In this chapter I give an overall picture of this class of materials as physicists see them. I provide an insight into their rich phenomenology and unusual behaviour (section 1.1), reveal those properties which hinder the classical approaches (section 1.2), and present the usual methods (section 1.3) and some new concepts (section 1.4) that physicists try to apply to describe them.

1.1 Playing in the Sand – Granular Experiments Granular materials are actually very simple: they consist of a great number of macroscopic particles interacting only with their neighbours. In case of dry samples the sole interactions are hard core repulsion and friction. However, in spite of their apparent simplicity, their behaviour can be very complex and often surprising (some of the nice reviews are: de Gennes, 1999; Jaeger and Nagel, 1992; Jaeger et al., 1996). In order to give an introduction to the diverse world of granular media, in this section we briefly summarize the different experiments performed with these materials. Here the presentation of the corresponding theories is restricted to a short explanation of the observed phenomena. The physical consequences of the unusual behaviour and possibilities of theoretical description are presented in the subsequent sections.

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G RANULAR M ATERIALS

(e)

(f)

Figure 1.1: Force chains in different experiments. (a-d) 2D array of birefringent pentagons. (a) The force pattern due to the grains’ own weight. (b) Combined response to gravity and a point force. (c,d) Net response to a point force for two different arrangements. (e) Force chains in a sheared sample. (Behringer et al., 2001) (f) Force chains in a simple pile. (Vanel et al., 1999)

1.1.1 Resting State The simplest possible experimental situation is when the material is at rest. Even at the first glance, the behaviour of granular samples is different from the other states of matter. They take the shape of the container holding them just like liquids, but they can maintain non-horizontal surfaces (up to a critical angle max ) and they can hold solid bodies on their surface. A more detailed investigation of these systems reveals further specific properties. As a first example let us consider the density of the packing, characterized by the packing fraction , i.e. the ratio of the volume of the grains and the bulk volume. In general the grains form a disordered structure, in which the weight of every grain has to be supported by its neighbours. Such stable configurations can be formed within a range of overall densities, depending on the construction history. The possible values of packing fraction depend strongly on the properties of the grains (size distribution, shape, etc.), its actual value is a key parameter determining the mechanical properties of the structure. As the simplest situation let us consider monodisperse, spherical grains. Experiments and computer simulations indicate that depending on the construction history, the packing fraction of a such a medium may vary between = 0:56 (“random loose packing”) and 5

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= 0:64 (“random close packing”). If < 0:56, then the formation of stable stress paths that are able to hold every grain is not possible. In principle higher densities than

= 0:64 could be achieved – an ordered hexagonal lattice corresponds to = 0:74 – but such high densities do not occur spontaneously (special filling techniques may lead to very high densities (Poliquen et al., 1997)). The reason is static friction among the grains: the particles constrain the motion of each other and the system freezes in a looser state. Another consequence of static friction is manifested in the magnitude of the pressure at the bottom of the sample. Unlike liquids, the pressure at the bottom of a granular column does not increase indefinitely if the height of the column is increased but exhibits an exponential saturation (Janssen, 1895; L.Vanel et al., 1998):

P = Pmax 1 e

h=h0 :

(1.1)

Above a critical height the weight of any excess material added to the top is supported by the walls via frictional forces. One of the most interesting properties of granular systems is that the grains only interact in the contact points, which form a random network inside the material. Any forces originating from the weight of the grains or from external effects can propagate only on this network (Eloy and Clément, 1997; Cates and Wittmer, 1999). This leads to a highly inhomogeneous distribution of internal stresses. The distribution of forces acting on single grains has been measured in a remarkably simple setup: a carbon paper was placed under the sample and the size of the marks made by the beads was analysed (Liu et al., 1995). The observed distribution showed an exponential decay at high forces, indicating that a relatively small part of the grains bears most of the total force. (These interesting features are demonstrated in an educational experiment that I assembled for the demonstrational laboratory course of modern physics (Tegzes, 2000). The experiment is described briefly in Appendix A.) The position of the grains bearing high forces can be visualized experimentally by using photoelastic grains, which change the polarization of transmitted light, if they are under stress. If such disks are viewed across crossed polarizers, then regions of high force will appear bright in a dark background. Furthermore, from a more detailed analysis of the interference patterns within a single grain the actual magnitude of the inter-particle forces can be determined. It turned out, that the grains bearing the largest forces are aligned along chain-like structures, with a typical width of 5 10 grain diameters, called 6

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Figure 1.2: The effect of construction history on the stress dip under a sandpile. Top: preparation by pouring from a point source. The stress dip is clearly present. Bottom: preparation by pouring through a sieve. The stress dip has disappeared. (Vanel et al., 1999) force chains (Dantu, 1967; Travers et al., 1986; Howell and Behringer, 1999; Vanel et al., 1999). The structure of the force chains has been examined under various conditions (see Fig. 1.1: from samples supporting only their own weight to systems bearing compressive, shear (Veje et al., 1998; Howell and Behringer, 1999) or point-like external forces (Rajchenbach, 2001; Behringer et al., 2001). A usual property of the arrangement of force chains is that forces of a given direction often propagate sidewards, resembling arches in architecture (Duran, 1998). A spectacular manifestation of such arching is reflected in the stress distribution at the bottom of a simple sandpile. Several experiments testified that right below the center of the pile, where the height is maximal, the pressure exhibits a local minimum (Hummel and Finnan, 1921; Jotaki and Moriyama, 1979; Brockbank et al., 1997). This phenomenon has also become a paradigm of memory effects, i.e., strong dependence on construction history, which is typical in granular media (Daerr and Douady, 1999a). If the sandpile is prepared by pouring grains to the top, then the pressure dip is clearly present (see Fig. 1.2). However, if the pile is prepared by a different method, e.g. by a homogenous rain of grains falling from a sieve, then the pressure dip is not observable (Vanel et al., 1999). The geometry of the two types of piles is very similar,

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but there are differences in the internal structure and it strongly influences the properties of the force chain network.

1.1.2 Sound Propagation Since the force chains determine the stress-propagating paths inside the granular medium, their existence has strong influence on the propagation of sound in these materials (Liu and Nagel, 1992). The transmitted sound is basically composed of two parts, a coherent ballistic wave travelling through an effective medium with a velocity in the order of

1000 m/s, followed closely by speckle-like multiply scattered waves reflecting the properties of the force chains. The first signal remains practically unchanged if the configuration of the packing is changed slightly, however, the second part is found to be non-reproducible, i.e. specific to the configuration at the level of individual contacts (Jia and Mills, 2001). A very small change in the configuration of the material can result in large changes in sound transmission. If the temperature of a single grain is raised by 1 K, meaning that its

size is increased by only 0:002 %, the change in the observed sound transmission can be reproducibly modified by as large as 25 %. If the heater is placed at a different position, the effect of heating may be different, sometimes it causes no discernible changes at all. The explanation may be the extreme heterogeneity of the material: if the heater lies on or near a force chain, it can have a much more dramatic effect on the local transmission of the sound wave than if it lies in a region away from any of the chains (Liu and Nagel, 1994).

1.1.3 Vibrational Excitations Due to internal dissipation motion of granular materials can only be maintained by a constant supply of energy. One of the most convenient forms of energy input is excitation by vertical vibration. The most important control parameter of vibrated granular systems under gravity is the ratio of the maximum acceleration of the vibrating base and the gravitational acceleration:

=

A !2 ; g

(1.2)

where A and ! are the amplitude and frequency of the vibration respectively. At low values the vibrations can play a role similar to an effective non-zero temperature, since they allow the system to travel in its phase space towards a configuration 8

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Figure 1.3: Oscillon. (Umbanhowar et al., 1996) of lower potential energy. In practice it means that the packing fraction of the material is slowly increased, and started from an inclined surface, the inclination angle is slowly decreased. Both of these relaxation processes were found to be logarithmically slow, i.e., the relevant parameters are proportional to log(t) over several orders of magnitude.

At higher vibration intensities ( > 1) the medium may display several different types of symmetry-breaking instabilities. The earliest observation of such an instability is due

to Faraday (1831), who first reported convection in vibrated granular systems. In the most common situation a pair of convection rolls is observed in a rectangular container: the material flows downwards in a thin layer near the container walls, which is compensated by a slow upward flow in the center (Ehrichs et al., 1995). This process builds a single heap at the center. In modified experiments convection in the opposite direction (Knight, 1997) and multiple convection rolls were also observed (Aoki et al., 1996). The experiments indicate that one single mechanism cannot explain all the observed features of convection. There is clear evidence, that wall effects play an important role: modifying the roughness or angle of the side walls has crucial effect on thew observed convection, e.g. it can reverse flow direction (Knight, 1997). However, convection was also observed in samples confined between two upright cylinders, where the sidewalls were not present at all (Pak and Behringer, 1993). In such arrangements the trapping of interstitial gas under the vibrated layer was proven to be essential: if the experiment was performed in vacuum or the base was perforated to prevent gas trapping, then no convection was observed (Pak et al., 1995). Horizontal shaking may also induce convection (Liffman et al., 1997).

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Figure 1.4: Various segregation mechanisms in granular materials. The black particles are smaller than the white ones. (a) Segregation pattern in a rotating drum at low rotation rates. The dark stripes are formed in successive avalanches. (b) The same experiment at larger rotation rates, where the flow of material is continuous. (c) Stratification. The material is confined between glass plates, grains are poured to the top. The drawing illustrates the sieving mechanism inside the avalanche which leads to stratification. (Gray and Tai, 1998) Further increasing the vibration intensity (

& 2:5) another class of instabilities oc-

curs in thin layers. In two-dimensional samples subharmonic standing waves can be observed (Douady et al., 1989; Clément et al., 1996) . In 3 dimensions these standing waves form complex patterns like stripes, squares or hexagons depending on the amplitude and frequency of the vibration (Melo et al., 1995; Bizon et al., 1998). The most famous of these patterns is a localized excitation called oscillon (Umbanhowar et al., 1996; Cerda et al., 1997) (Fig. 1.3).

1.1.4 Segregation Effects A prominent phenomenon of granular materials is that mixtures of particles with different properties tend to segregate. The simplest example is the famous “brazil nut effect” (Mobius et al., 2001), in which the larger particles of a shaken mixture spontaneously raise to the top surface. But segregation occurs in practically every experimental situation, where granular mixtures are in motion (Fan et al., 1990; Rosato et al., 1991). The most important experiments in which segregation has been investigated include vibrated granular samples (Ahmad and Smalley, 1973; Duran et al., 1993; Jullien et al., 1993; Vanel et al., 1997; Shinbrot and Muzzio, 1998), quasi-2dimensional rotating drums (Baumann et al., 1995; Gray and Tai, 1998), where segregation occurs in the radial direction, and long rotated cylinders, which exhibit both radial and axial segregation (Hill and Kakalios, 1994;

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Zik et al., 1994; Nakagawa et al., 1998). Some interesting patterns are presented in Fig. 1.4. An especially interesting phenomenon occurs when a mixture of grains of different sizes is poured onto a pile between two parallel plates. Depending on the properties of the grains either simple segregation is observed (large grains are accumulate at the bottom of the pile), or a new phenomenon: the mixture spontaneously stratifies into alternating layers of large and small grains (Makse et al., 1997; Gray and Tai, 1998).

1.1.5 Deformations and Flows A granular property of great technological importance is the ability of these materials to deform and flow. As first described by Reynolds, a densely packed granular material has to expand in order to deform, so that the grains can move around each other (Vermeer and de Borst, 1984). One can observe this dilatancy when walking on the beach. The wet sand around one’s foot dries out as weight is put on it. The deformation caused by the pressure of the foot causes the sand to expand and allows water to drain away (Jaeger et al., 1996). On the other hand, loose packings may be compressed when undergoing deformations. At an appropriate packing fraction, the material can be deformed without volume changes, this is the so-called critical state. In many cases if a sample is deformed continuously, then it dilates (compacts) spontaneously to the critical state, and then it remains there — the system evolves along the “critical state line” in the phase space. 1 In slowly deformed materials the arrangement of the grains is changing continuously. Since the network of force chains is very sensitive to rearrangements of the packing, one can observe strong temporal fluctuations in the measured stress (Veje et al., 1998). Shear experiments in a Couette geometry revealed that the packing fraction is a crucial parameter in determining the properties of the force chain network. For loose packings one can observe intermittent, long radial chains, while denser systems display a tangled dense network. A continuous transition is observed between the two types of behaviour (Howell and Behringer, 1999). Another important phenomenon is that under continuous shearing the material does not deform homogenously. The dilation and displacement of grains is typically limited 1

The expression “critical state” is also used for systems close to a continuous phase transition. These two concepts are not related. The fact that continuously deformed systems spontaneously reach critical state in the soil-mechanical sense is NOT self-organized criticality (see Sec. 1.4.1).

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10 grains wide called a shear band (Mueth et al., 2000; Vermeer

At higher velocities and under smaller stresses granular materials are fluidized, and can flow similarly to fluids. Sand can be poured from a container to another. However, there are also fundamental differences between granular and fluid flows. One of the most important differences is that granular flows are often confined to a thin surface layer of the material (see Chapter 2 for details about surface flows). Bulk flows also exhibit specifically granular properties, the most famous example is the emergence of travelling density waves in granular materials flowing through a hopper (Baxter et al., 1989). The existence and the properties of the waves depends strongly on particle shape. Travelling density waves can also be observed, when a granular material flowing through a vertical pipe is stopped suddenly (Moriyama et al., 1998)

1.1.6 Clustering in Granular Gases If sufficiently strong vibrations are applied, granular materials can also display a loose, gas-like state (Olafsen and Urbach, 1998), especially if the experiment is performed in low gravity (Falcon et al., 1999). An alternative method to reach such a state is to use spherical particles rolling on a very smooth horizontal surface, and drive them by the vibration of one of the vertical walls (Kudrolli and Gollub, 1997). In all cases the inelastic collisions between the particles lead to an instability of the system: the particles tend to form coherently moving clusters (Goldhirsch and Zanetti, 1993). This instability has profound theoretical consequences (Du et al., 1995; Zhou, 1998). Space limitation doesn’t allow us to mention all the interesting phenomena in granular materials. A great number of other recent experiments have been published which certainly deserve attention (Medina et al., 1995; Alonso and Herrmann, 1996; Duran et al., 1996; Scherer et al., 1996; Menon and Durian, 1997; Betat et al., 1999; Amarouchene et al., 2001).

1.2 Difficulties of Classical Approaches Given the extreme diversity of the behaviour of granular materials, it is a challenging task to explain them theoretically. Moreover some properties of granular materials — the absence of thermal fluctuations and the dissipative interactions — cause serious difficulties for the well established theories of classical physics. In this section we examine the very 12

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fundamental principles of statistical mechanics, hydrodynamics and continuum theory, and show why they are incompatible with granular materials.

1.2.1 Statistical Mechanics The traditional approach of statistical mechanics (Nagy, 1971) considers two different levels of description of a system of many particles. A given macro state is characterized by the values of the measurable quantities (e.g. temperature, pressure, volume, mass etc. if we consider some gas). On the other hand a micro state corresponds to a fixed set of values of all the microscopic parameters of the system (e.g. the position and momentum of all the molecules). Clearly, a lot of different micro states may lead to the same macro state. Then two important assumptions are made: (i) in a closed system all the micro states have equal probabilities, and (ii) in sufficiently long time all the micro states are realized (ergodic hypothesis). Thus the probability of a macro state is given by the number of micro states that correspond to it; we observe the one with the greatest number of micro states. To describe the number of states corresponding to a macro state, the notion of entropy is introduced: S = kB ln , where kB is Boltzman’s constant and is the number of micro states2 . This way the macro state corresponding to the greatest number of micro states is marked by the maximum of entropy. The second law of thermodynamics has now a plausible explanation: a closed system develops in the direction of macro states with greater number of microstates, thus entropy cannot decrease. The most fundamental property of granular matter which makes classical statistical physics fail, is that the grains are macroscopic and so thermal energy is negligible compared to the gravitational energy of the particles (kB T = mgd 10 10 , where m and d

are the mass and diameter of a particle, g is the gravitational acceleration). Consequently the system is very far from being ergodic, it is frozen in a single micro state. The number of micro states corresponding to a macro state is thus not relevant in granular systems. In case of slow changes the system does not follow a path of successive equilibrium states, but only a succession of specific microstates. Instead of the classical thermodynamical driving forces the system is driven by microscopic mechanics, and thus entropy may actually decrease. The well known general definition is S = k B state. In a closed system pi = const = 1= , and thus S 2

13

Pp

i i = kB

ln pi , where pi is the probability of the i th ln .

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Figure 1.5: Clustering instability in a 2D granular gas simulation with no external driving. A classical example of irreversible thermodynamic processes is the mixing of particles. It is easy to show that in ideal gases the maximum entropy principle prefers mixed states.3 This leads to the commonly observed phenomenon, that mixed gases never demix spontaneously. In contrast, since granular materials don’t explore their phase space, the entropy may be decreased. As discussed earlier in Section 1.1.4, in excited granular mixtures the typical situation is not mixing, but segregation. We have so far spoken about systems that are at least close to a resting state. However, there exist situations, where the grains are in a gas-like, fluidized state. In such cases the above mentioned lack of ergodicity does not seem to appear. However, the standard ways of descriptions are hindered by another property of the system: the dissipative interactions. Frictional forces and inelastic collisions in the system continuously dissipate the kinetic energy in the bulk of the system. This property cannot be compensated by energy input at the boundaries, in large enough systems the standard solutions will become linearly unstable towards clustering (Fig. 1.5). The source of instability is clear: at denser regions of the system the number of collisions and thus the rate of energy dissipation is 3

Let us consider a closed container consisting of two equal compartments which contain different types of ideal gases. Let us denote the number of particles on each side by N 1 and N2 , and the number of states available for the two types of particles by 1 and 2 . The entropy of the system is equal to S = N N kB ln( 1 1 2 2 ) = kB N1 ln 1 + kB N2 ln 2 . If then the wall between the compartments is removed, then the number of states is doubled for all the particles, and the entropy reads S 0 = kB N1 ln(2 1 ) + kB N2 ln(2 2 ). We can write this as S 0 = S + Smix , where Smix = kB (N1 + N2 ) ln 2 is the mixing entropy, which is always positive.

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increased. Thus incoming particles are slowed down and captured, further increasing the local density. A special and interesting case of clustering is when the density of the materials becomes so large, that an infinite number of collisions occur in finite time, meaning that the velocity of neighbouring particles becomes exactly the same, this phenomenon is called inelastic collapse. The same phenomenon occurs when an inelastic ball bouncing on the ground comes at rest. The crucial difference is that in the granular sample there is no attractive force between the particles, the infinite number of collisions is induced by the many-particle dynamics. In general, it is an inherent property of granular materials, that self-organized processes drive drive them away from a levelled, homogenous state. Besides segregation and clustering we can mention a lot of other phenomena: e.g. the density waves in flowing granular materials, or the heaping and the various exotic patterns exhibited by vibrated granular beds. In this context it may even be surprising, that in some cases thermodynamic arguments may be successfully applied to granular materials. Section 1.3.4 gives an overview of these possibilities.

1.2.2 Hydrodynamics Sand can be poured almost like liquids, thus it is a natural idea to describe granular flows using the powerful formalism developed for liquids. The fundamental equation of hydrodynamics is the well known Navier Stokes equation:

@v + (v r)v = F @t

rP + 4v;

(1.3)

where is the density of the liquid, v is the local velocity, F represents volumetric forces

(e.g. F = g in case of gravity), P is the pressure, is the viscosity. r means gradient, 4 is the Laplacian operator. Though students meeting it the first time may find it complex, its derivation requires remarkably few ingredients. Apart from the simple assumptions, that normal stresses are isotropic:

xx = yy = zz ;

(1.4)

and shear stresses are proportional to the rate of deformation:

xy = 2 _xy ; : : : ; 15

(1.5)

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only momentum conservation and a clear thinking is needed. Equation (1.3) is in fact a P F . The left hand side describes the change continuum version of Newton’s law: ma = in local velocity via local acceleration and due to convection: material of different velocity arrives from the neighbourhood. The right hand side summarizes the acting forces: gravity, pressure differences, and internal friction. The lack of specific assumptions in (1.3) enables it to describe a wide range of phenomena, and it gives a general description of newtonian liquids. However, if we try to apply eqn. (1.3) for granular materials, we soon meet difficulties. Of course, dissipation mechanisms in granular materials cannot be described by (1.5). Depending on the velocity it may be dominated either by inelastic collisions or frictional forces. But (1.4) and (1.5) can be replaced by equations describing internal forces in granular samples. These will add some new terms to the right hand side of the Navier Stokes equation (1.3), which now would be a general description of granular materials. Unfortunately, this scenario does not always work. The most fundamental problem lies right in the concept of describing these materials with differential equations. Such approaches always assume that the quantities (e.g. velocity) are smooth (i.e. differentiable) functions of space. This is a reasonable assumption for liquids, since even the smallest relevant lengthscales of the observed flow include millions of microscopic molecules, constantly moving and interacting due to thermal motion, and eliminating all discontinuities. But granular materials lack both the separation of lengthscales and the microscopic thermal velocity scale. The only velocity comes from the flow of material, and the velocity field is not necessarily smooth. It is well possible that a grain moves quite quickly, why its nearest neighbour is at rest. This makes the application of differential equations questionable. The practical solution for that problem is that regions of different behaviour are separated and described separately. Recent models describing surface flows give an elegant example: they write coupled equations for the static bottom and the flowing layer (see sec. 1.3.5). A liquid at rest is quite similar to a flowing liquid - the thermal motion of the molecules are the same. For this reason the Navier Stokes equation applies for practically all experimental situations: it can describe pressure in a lake 4 as well as flow in a cylindrical Substituting v 0 to (1.3) and assuming that the only acting force is gravity yields 0 = g rP . The solution is straightforward and we gain the formula of hydrostatic pressure: P = gh (h denotes depth). 4

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pipe5 or the drag force acting on an object.6 On the other hand different flow types and the resting state are qualitatively different for granular materials and require different models of description. Quick, and relatively dense collisional flows are often described by the modified Navier Stokes equation, regions of different type (being at rest, undergoing quasistatic deformations or possibly behaving like a loose gas) are handled with different methods. Constructing a general model that would be able to describe the different phenomena in a single framework similarly to eqn. (1.3) would be a historical step in our understanding of granular materials. Unfortunately such an ultimate model is far from being realized at the moment. As an alternative, the rapidly developing computer simulations (see Sec. 1.3.7) solving equations of motion for each particle may take the role of such a universal tool.

1.2.3 Classical Continuum Description Classical continuum description faces similar problems. While not debating the usefulness and efficiency of these models — many buildings and earth structures prove their abilities — we mention some of their theoretical constraints that are of physical interest. Again, the first problem arises from the macroscopic size of grains. These models usually consider a stress tensor ij (r; t) describing the internal forces, and a strain tensor ij (r; t) corresponding to the deformation of the material. The material is then described by the constitutive equation, which has the form

ij (r; t) = f (ij (r; t); _ij (r; t); :::):

(1.6)

In order to define ij , a small piece of material with linear sizes of Æx1 , Æx2 , and Æx3 is considered. Then the local stresses are described as the forces acting on the faces of this small piece of material divided by the area of the faces. Here Æxi should be much smaller than the typical length describing the variation of any acting force, but still larger by several orders of magnitude than the size of the particles — this ensures that the definition of the stresses does not depend on the choice of Æx i . If the material in question 5

We assume a constant velocity in time and in the direction parallel to the pipe and neglect volumetric forces. Then we solve the Navier-Stokes equation in cylindrical coordinates, which leads to the well known quadratic velocity profile: v (r) = P24lP1 (R2 r2 ) (here R and l denotes the sugar and length of the pipe, r denotes the distance from its axis, P 1 and P2 denote pressures at the two ends). 6 We consider F 0, and assume a small and stationery velocity so that the left hand side of (1.3) can be neglected. The resulting equation is rP = 4v . Solving it for a spherical obstacle yields for the total force acting on the obstacle F = 6du, where u is the velocity of the liquid far away from the sphere.

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is an ordinary solid body, where the particles are atoms, finding such a length scale means no problem. However, for granular materials the length scale of the variation of the force is often similar to the particle size, which makes it impossible to choose an appropriate Æx. If we consider very large granular samples with smoothly varying external forces, then we may attempt to average the internal forces over a great number of particles. However, such an averaged description will have important limitations. (i) It completely neglects the large force fluctuations due to the formation of force chains. (ii) Strict mathematical analysis reveals, that the stress and strain tensors resulting from such an averaging process do not correspond to each other (Bagi, 1999b). The deformation corresponds to the motion of the grains themselves (in a mathematical description the so called material cells surrounding them), while the forces are acting at the contact points and thus they are determined by the network of contacts (and the corresponding space cells) (Bagi, 1999a). The material cells and the space cells never coincide, so the application of constitutive equations like eqn. (1.6) is questionable.

1.3 Directions of Granular Physics Since the work that I present in this thesis is mainly experimental, I am far from being competent to give a comprehensive overview of granular theories. However, this introductory chapter would be incomplete without presenting at least a subjective selection of the possible approaches.

1.3.1 Phenomenological Models The first level of description aims the understanding of single phenomena, the explanation of “what is happening” in an experiment, qualitative or possibly quantitative reproduction of the observed features or measured quantities. Both the investigated quantities and the equations are strongly related to the specific experimental situation. The model quantities are usually the measurable quantities, the interaction between them is the simplest possible form that can explain the given phenomenon. The microscopic dynamics is handled only qualitatively to justify the equations, plausible pictures, analogies are often used. Though these models have only a limited predicting power to other situations, these are usually the ones, that non-specialist remember about a given phenomenon. Also due to their simplicity these models have an important role in establishing links with other 18

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fields of science. Given the rich phenomenology of granular materials such models are abundant too, we can mention plausible models explaining segregation and stratification mechanisms, the simplest heaping models, the “parking lot model” for slow relaxations, or the famous q-model for force chains.

1.3.2 Characterization of Materials A completely different approach is motivated by industrial applications. They need to predict the rough behaviour of specific materials. They are only interested in the simplest properties like stability or average forces arising at deformations, but the materials themselves can be quite exotic, e.g. shavings, agricultural produces, or simply a soil of a given composition. In this context the most important need is to determine appropriate quantities that capture essential features of the properties of the material and thus can be used to predict the behaviour. Then usually standard test procedures are defined to measure these quantities. A related approach is when the characterization of a given state is attempted. It is well known, that the qualitative nature of liquid flows is determined by a single dimensionless quantity, the Reynolds number. Similar dimensionless parameters describing the relative importance of different effects can be defined for granular systems too (Nase et al., 2001).

1.3.3 Continuum theories Though continuum models completely ignore the granular structure and sometimes yield wrong predictions, the several decades of intensive research in this field make them a very powerful tool, and in case of very large samples they are often correct. Actually, these are the most widely used in engineering practice. If discrepancies between the models and real behaviour are revealed, then some extension of the model can often reduce the contradiction. Well-known examples are the change of associated flow rules to nonassociated ones to account for the dilatancy effects in shear bands (Vermeer and de Borst, 1984), or the application of Cosserat or higher order continua to describe phenomena related to the rotation of particles.

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1.3.4 Thermodynamical principles In spite of the fact that granular materials do not explore their phase space, thermodynamical principles may be applied in some cases. For example the size distribution of natural soil particles, and the distribution of contact forces inside a granular medium (Bagi, 1997) can be well approximated using the principle of maximum entropy. The explanation may be that both distributions arise from stochastic processes involving a lot of random events. These events are the crushing of individual soil particles into random fractions, and the quasi-random distribution of the forces acting on a grain to its neighbours. The different outcomes of these random events take the role of micro states here, the overall distribution corresponds to the macro state. Ergodic time averaging is substituted by averaging over the great number of random events. The maximum entropy principle simply selects the most probable outcome of the processes. Another class of models deals with situations, where the grains are in motion due to vibrations or a simple flow. In both cases some random velocities are generated in addition to the average velocity of the grains. These random motions can be regarded as a “granular temperature”: T =< v 2 > < v >2 : (1.7) Though a full analogy with ordinary thermodynamics cannot be achieved, this granular temperature is responsible for many observed phenomena like particle diffusion, pressure, and the transport of momentum and energy. An alternative description completely dismisses the kinetic definition of temperature and replaces it by compactivity, while role of energy is taken by the volume of the system (Mehta and Edwards, 1989). Using this approach the complete thermodynamical formalism can be applied. Interestingly in this framework the segregation of particles is a process that increases the generalized entropy, so it can be described as a phase transition.

1.3.5 Granular flow models Several models have been proposed to describe granular flows. We have already mentioned that the way of description depends on the shear rate, the possible approaches include plasticity theories for slow deformations (Vermeer and de Borst, 1984), stochastic cellular automaton models for bulk flows (Caram and Hong, 1991) and kinetic theories

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for completely fluidized layers (Jenkins and Richman, 1985). In the description of surface flows an original and very fruitful idea was to handle the static and flowing part together, by writing two coupled equations for them (Bouchaud, Cates, Prakash, and Edwards, 1994,Bouchaud et al., 1995). In the original BCRE model the pair of equations looks as follows:

R_ (x; t) = h_ (x; t) =

@x ( vR ) + @x (D@x R ) + (h; R) (h; R):

(1.8) (1.9)

Here h(x; t) is the height of the grains at rest, R(x; t) is the local density of rolling grains, v is the velocity, D is the diffusion constant. Eqn. (1.8) contains 3 terms, the first is convective, the second is diffusive, accounts for the transition of immobile grains to rolling grains and vica versa. The coupling term was chosen so that it satisfied a few physical requirements based on considerations about the collisions of the grains on the fixed base. Several modified versions of this model were proposed (de Gennes, 1995; Boutreux et al., 1998; Prigozhin and Zaltzman, 2001) In an alternative approach the starting point consists of simple conservation laws, the Saint-Venant equations (Douady et al., 1999; Bonamy et al., 2001), and the coupling is related to the effective force acting on the flowing layer:

@t ( h + R ) + @x ( Rhvxi ) = 0; @t ( Rhvxi ) + @x ( Rhvx2 i ) = F:

(1.10) (1.11)

Here Rhvx i is the flow rate, Rhvx2 i is the x-component of the flux of momentum, F is the force acting on the flowing layer. Eqs. (1.10) and (1.11) ensure conservation of mass and momentum respectively. The missing relations hv x i(R; h), hvx2 i(R; h) and F (R; h) are determined by measurements. Both models show good agreement with experiments.

1.3.6 Microscopical theories There exist really strict approaches that are formulated on the level of the grains and are handled with mathematical accuracy. Typical examples are models of granular contact topology (Goddard, 1998; Bagi, 1999a) or the kinetic granular models. This level of description becomes rather complicated when applied to complex experimental situations. The solution would be to apply appropriate approximations relevant to different situations and derive simpler macroscopic models from the microscopic one, similarly to the 21

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derivation of classical physics from quantum mechanics. However, as described in the previous chapter, this step is extremely difficult sometimes impossible for granular materials. For example, tessellation theory does not lead to continuum mechanics in the limit of large samples(Bagi, 1999b).

1.3.7 Computer Simulations Computer simulations represent a special way of description of physical systems. Since the appearance of the first DEM studies the tremendous increase in the computational power of computers dramatically expanded the possibilities of simulation methods, and they became widely applied in various fields. When applied for granular materials, a common feature of these techniques is that they handle the equations of motion of particles separately. All the other properties may vary. The situation described can be either static or dynamic, the particle shape can be spherical rough or possibly fragile, the motion can happen on- or off-lattice, the equations governing the collision may vary from simple rules of cellular automatons to sophisticated, almost realistic collision models, the numerical method for solving the model can be event driven or use a given timestep, etc. Nowadays computer simulations are able to reproduce any exotic experimental observations within a few months — in this sense they can be regarded as a general theory. Their contribution to the understanding of a phenomenon is mainly a separation of essential and non-essential parameters, that are required/not required for the reproduction of the experiment. On the other hand they can be regarded as special types of experiments. Since they provide coordinates, velocity and angular velocity components for all the particles they can be used to gain a more detailed picture of the dynamics. If the measurable quantities are reproduced, then the model system can be assumed to be similar to the real one, and we can easily extract further information about quantities which are hard to measure experimentally.

1.4 Novel Concepts and Fruitful Analogies Granular materials are simple in the sense that they are easy to imagine. On the other hand they exhibit complex phenomena, and their physics is highly unusual. The combination of these two features make granular materials an ideal candidate to draw analogies with other physical systems. In the following we present a few examples.

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1.4.1 Self-Organized Criticality If a grain is placed on the top of a sandpile, then it may either find an equilibrium position and stop there, or it may dislocate other grains and initiate an avalanche. An avalanche s can in principle be very small, influencing only the immediate neighbourhood of the starting grain, but it can also be very large, spanning the whole system. If the slope of the pile is small, then usually small avalanches occur. But if we continue to add grains, then the slope increases and large avalanches will appear more and more frequently, until they are able to balance the addition of the grains. This picture of the dynamics of avalanches on a sandpile gives a plausible explanation for large fluctuations in other systems. Based on this idea, Bak, Tang and Wiesenfeld (1987,1988) introduced a simple sandpile model to describe spatially extended driven dynamical systems. The sandpile in the model consists of grain columns on a lattice, and grains are added one by one to different places of the lattice. Whenever the height difference between neighbouring columns exceed a critical value, a grain topples from the higher column, this in turn may induce further instabilities. In the steady state the probability distribution of the avalanche sizes s

follows, P (s) / 1=s, and all the relevant quantities exhibit power law distributions. This behaviour is analogous to equilibrium systems near a critical point. The fundamental difference is that in the sandpile model the system spontaneously evolves to the critical

state without an external fine-tuning of the control parameter. In this sense the sandpile is self-organized critical (SOC). Self-organized criticality has been suggested to describe many different kinds of dynamic problems (Nagel, 1992). Thus 1=f noise in metal wires, the power laws seen in turbulent systems, the power-law distribution of earthquake sizes, the fractal nature of coastlines, the power-law distribution of mass in the universe, the distributions observed in the cellular automaton game of life have all been ascribed to some self-organized behaviour. Clearly, if even a small portion of these systems turns out to be self-organized critical, the model is important to understand. This is true in spite of the fact that experiments indicated, that the original system that inspired the model — avalanches on a sandpile — generally does not exhibit SOC (see Sec. 2.3.1 for exceptions and details).

1.4.2 Slow Relaxations Logarithmically slow relaxations in granular materials strongly resemble the behaviour of a great variety of systems, like glasses and spin glasses, amorphous ferromagnets, fer-

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rofluids, high-temperature superconductors and dipolar-coupled arrays (see Jaeger and Nagel, 1992 and references therein). Indeed, the underlying physics is similar: these systems usually have a frustrated structure, where the flip of a spin or motion of a molecule is constrained by the neighbouring region. Finding the minimal energy state is thus extremely difficult. The system is locked into a local minimum, and reaching a lower energy state requires a synchronized change of several particles. This scenario is analogous to the interlocking of the grains and the formation of jammed states in granular samples. There too, at high densities the further increase of packing fraction requires the collective motion of many particles. A simple corresponding picture is a parking lot full of randomly positioned cars. The higher the density the more cars have to move in order to make room for an extra vehicle (Jaeger et al., 1996). If the displacement of grains are independent random event, then the probability of a cooperative motion of particles decreases exponentially with the number of grains involved. Thus the densification rate decreases exponentially in time, which means that the approach to steady state is logarithmic in time (Ben-Naim et al., 1996).

1.4.3 Chaos Chaos theorem (Ott, 1993) is usually used for much simpler dynamical systems than granular materials. Chaotic systems are strictly deterministic, the noise-like behaviour is only a consequence of the nonlinear dynamics leading to quick divergence of neighbouring trajectories and sensitivity to initial conditions. In contrast, granular dynamics in general can be characterized by a lot of stochastic elements. However, there are some simple situations, where granular materials behave in a deterministic way and chaos theorem is applicable. An example is the effect of vertical vibrations. If a single inelastic particle is bouncing on a vibrating plate, its dynamics is very close to chaotic: as the frequency is increased it undergoes a series of bifurcations until it reaches the self-reanimating chaotic state (Kowalik et al., 1987). If a granular sample of many particles is vibrated at low frequencies, then it behaves quite similarly: it moves like a single block. The transition to pattern formation at higher frequencies is then closely related to the period-doubling bifurcations (Melo et al., 1995). The patterns diminish, when the chaotic state is reached. A related system, in which an inelastic ball is bouncing on a rough inclined plane also shows chaos (Valance and Bideau, 1998).

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A different example is the avalanche flow of granular materials in rotating drums, which is very regular and can be treated by simple mathematical models. (Khakhar et al., 1999) show computational evidence for chaos in the mixing process of granular materials in non-circular tumbling mixers.

1.4.4 Traffic Models A street full of pedestrians or vehicles can be regarded as a system of many particles. Naturally in general the forces driving the “particles” and the interactions between them are extremely complex — they are all individual persons with their aims, plans way of thinking, habits, friends, enemies, etc. However, there are many situations, when the behaviour of the individuals is much simpler. For example drivers of cars on a highway, spectators in a stadium after the end of a match or panicking people in a burning building all want practically the same, and behave similarly. In such situations the important properties of the system can be well described by molecular dynamic, kinetic or fluid-dynamic approaches, with suitable simple interaction laws between the particles (Helbing et al., 2000b; Helbing et al., 2000a) — these approaches are usually called “traffic models”. The similarity between granular flows and traffic models is apparent: discrete particles are in motion with dissipative interactions between them (people always slow down if they get too close to each other). Consequently, the equations of traffic models are very similar to lattice models of granular flows. Indeed, the cars moving on the highway are quite similar to a granular medium flowing in a pipe. But similarities are also found on a deeper phenomenological level: formation of clusters and density waves, segregation and stratification phenomena, as well as oscillatory flows are not only observed in granular, but also in traffic systems (Helbing, 1998).

1.4.5 Granular Ratchets Finally we present a much less widely known analogy. Many kinds of biological motion, ranging from transport mechanisms inside a cell to human muscle contraction, are driven by molecular combustion motors. A remarkable property of these processes that they do not involve gradients of external fields or chemical potentials that extend over the distance travelled by the moving objects. Instead, macroscopic motion is induced on the basis of purely local effects, the necessary ingredients are vectorial asymmetry, asymmetry to time reversal and both thermal and non-equilibrium fluctuations.

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To give an example: in human muscles myosin heads move along actin filaments. The actin filament provides a periodic asymmetric potential, temporal asymmetry is provided by viscous damping, and non-equilibrium fluctuations arise from regulated ATP hydrolysis which induces conformation changes in the myosin heads. As a consequence the myosin molecule alternates between phases when it binds to the filament and when is it released and performs Brownian motion. When the head is rebinding to the filament it does not necessarily return to its previous place, it can bind to neighbouring binding sites. Due to the local asymmetry in the system the neighbour in one direction is preferred, which leads to net motion. In the corresponding theoretical models, known as thermal ratchets, fluctuationdriven transport phenomena can be interpreted in terms of overdamped Brownian particles moving through a periodic but asymmetric, one-dimensional potential in the presence of non-equilibrium fluctuations. Typically, a sawtooth-shaped potential is considered, and the nonlinear fluctuations are represented either by additional random forces or by switching between two different potentials. In vibrated granular samples all the necessary ingredients of the ratchet mechanism are present: fluctuations arise from the vibration and particle collisions, spatial asymmetry is induced by gravity, temporal asymmetry is due to inelasticity. Indeed, explanations of granular convection that are based on wall friction or internal shear usually consider a ratchet-like mechanism (Aoki et al., 1996; Knight, 1997). In order to further explore this analogy our group has carried out experimental and simulational studies to investigate the flow of granular materials in a vertically vibrated system whose base has a sawtooth-shaped profile (Tegzes, 1998; Derényi, Tegzes, and Vicsek, 1998; Farkas, Tegzes, Vukics, and Vicsek, 1999). We found that a ratchet mechanism induces a net horizontal flow in the system. This flow exhibits novel collective behaviour, both as a function of the number of layers of particles and the driving frequency. In particular, under certain conditions, increasing the layer thickness leads to a reversal of the current, while the onset of transport as a function of frequency occurs gradually in a manner reminiscent of a phase transition. Similar collective effects were reported in ordinary thermal ratchets.

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Chapter 2 Continuous and Stick-Slip Motion in Granular Media 2.1 Stick-Slip Processes in Nature 2.1.1 Definition and Examples It frequently happens in nature that a system under continuous driving force responds in an intermittent way. Time intervals when the system is at rest and potential energy is accumulated alternate with active periods, when the system relaxes and potential energy is decreased. Since the simplest interaction leading to such behaviour is friction between two moving objects, these phenomena are called stick-slip processes. The observed fluctuations in the potential energy of the system may be of two different types. In regular stick-slip processes there is a well-defined threshold, where the relaxation begins and and the downward step during the slip process is always of the same size. Thus the variation of potential energy is a periodic sawtooth signal. This is the traditional case. But in more complicated systems both the threshold and the magnitude of relaxation may depend on internal parameters and change in time. We thus observe an irregular stick-slip process (Rabinowicz, 1965) with a random-like signal with both small and large steps (Fig. 2.1). One does not have to create sophisticated experimental conditions to observe stickslip motion. Starting from the frictional example: if we push heavy furniture, most probably it will proceed in discrete steps. Also, stick-slip motion is responsible for most of the sounds generated by friction from the squeaking of an old door to the sound of the violin.

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AND

S TICK -S LIP M OTION

IN

G RANULAR M EDIA

Force (N)

Periodic

Random

Stepped

Time (s)

Figure 2.1: Different types of stick-slip motion. The data describes fluctuations in the granular drag force (Albert et al., 2000). The granular drag experiment is presented in Chapter 3. Stick-slip motion has been observed in various frictional experiments from solid-onsolid friction of rough surfaces (Heslot et al., 1994) to delicate studies of lubricated films between smooth mica surfaces (Gee et al., 1990). Since one of the dominating interactions in granular materials is friction among the grains, it is not surprising that dense granular materials exhibit various stick-slip phenomena (described in detail in section 2.1.4). However, there are many other processes (many of them not related to friction) which also lead to stick-slip like behaviour. A simple example is the continuous charging of two metal balls close to each other, which exhibit discrete discharge events in the form of small sparks. Another well-known phenomenon is the sudden occurrence of earthquakes due to a slow continuous motion of tectonic plates. A more sophisticated example is the intermittent collapse of macroscopic superconducting regions due to a continuously increased parallel magnetic field in ultrathin granular superconducting films. The internal dynamics of foams (like a simple shaving foam) also has stick-slip features: macroscopically homogenous shear deformation leads to a series of discrete microscopic bubble rearrangements. Finally, I think it is not far-fetched to say that stick-slip fluctuations are observable in human behaviour too. A bit frivolous, but I think relevant example is that the slowly increasing pressure to finish a thesis relaxes in intermittent efforts, when the work proceeds quickly, separated by less active periods.

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ks

IN

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vs

M

Figure 2.2: Sketch of the spring-block system of the minimal stick-slip model. In the following subsections we analyse various models describing stick-slip motion. Here we present a bit more detail of the calculations than usually for two reasons. Firstly, the literature we have read on the topic contained some inaccurate or unclear statements: here we make an attempt to clarify these. Secondly, we will apply a similar model to interpret our experimental results in Chapter 5.

2.1.2 Minimal Model The simplest model system to study stick-slip motion consists of a block with mass M

sliding on a flat, frictional surface, and pulled by a spring, with a spring constant k s (Fig. 2.2). The “free” end of the spring moves with a constant velocity v s . As a first approximation let us use the Coulomb friction law learned in high school. If the material is sliding, then the frictional force is constant independently of the velocity:

Fsliding friction = sl Fn =: Ff ;

(2.1)

where Fn is the normal force pressing the sliding objects to each other, sl is the coefficient of sliding friction. If, however the material is at rest, then static friction cancels the net tangential force Ft , provided that this is smaller than a critical value:

Fstatic friction st Fn =: Fmax ;

(2.2)

where st is the static friction coefficient. In order to achieve stick-slip motion the only thing we have to assume is that the maximum value of static frictional force Fmax is larger than the force in sliding friction Ff . Let x(t) be the position coordinate of the sliding block. Then the equation of motion

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x_ = 0 and ks (vs t x) < Fmax M x = k (v t 0x) F ifotherwise. s s f

(2.3)

Clearly, this model has a steady state solution:

xS:S:(t) = Ff =ks + vs t

(2.4)

Now we’ll show that it also has a stick-slip solution. For simplicity we first consider the case when the block is pulled very slowly. In the initial state let the block be at rest and the spring be relaxed. The block remains at rest — stays in the “stick” phase — until the force in the spring reaches Fmax . At that point the block starts to slide, and the frictional force is suddenly reduced to Ff . The block performs a half period of harmonic oscillation around xS:S: , then it stops — this was the “slip” event. Here the process starts over and stick and slip events continue to alternate.

The dynamics of the minimal model system at very low velocities is plotted in Fig. 2.3, see the continuous lines represent the low-velocity limit. For v s 0 the slip event is represented by a half circle in the y_ (y ) space (Rajchenbach, 1990).

It is straightforward to solve this model even without the assumption that v s is small. Let us introduce a new variable: y (t) = x(t) xS:S: (t). Then (2.3) is reduced to:

M y =

0 if y_ = vs and ks y otherwise,

y < yslip

(2.5)

where yslip = (Fmax Ff )=ks . The coordinate system has been changed: now we have a fixed spring with an attached block, and the flat surface underneath is moving backwards with a velocity vs . If we assume that t dynamics of the slip is described by:

= 0 at the beginning of the slip process, then the

y (t) = A cos(!st ) where !s

(2.6)

p

= ks =M , A and are constants. The initial conditions y (0) = y slip and y_ (0) = vs require v = arctan( s ) (2.7) !syslip y A = slip (2.8) cos We omitted the x_ < happens. 1

0 case for simplicity.

If the spring is not compressed initially, then it never

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a

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x' x

s tic k

b

y

c

τ'

y'+vs (=x')

d

ystop s tic k

0 ystart τ

0

τ < τ'

ystop

ystart 0

t

t

y

Time between slips (T)

Slip duration ( τslip )

Figure 2.3: Behaviour of the minimal stick-slip model. (a) The frictional force as a function of velocity. (b) Time evolution of the position x(t) of the block. Solid line corresponds to vs 0, dashed line represents a higher velocity. The straight lines show the steady state position x S:S: . (c) y (t) = x(t) xS:S: for the same velocities. (d) The velocity of the block as a function of its position (x_ (y )) calculated from the same data.

2π/ωs

π/ωs

0

0.01

0.1

1

10

100

Dimensionless Driving Velocity ( vs / ωsyslip )

1E-3

0.01

0.1

1

10

100

Dimensionless Driving Velocity ( vs / ωsyslip )

Figure 2.4: Characteristic times of the minimal model as a function of driving velocity. a Slip duration slip , (b) time lapse between successive slip events. This solution holds, until the block stops, i.e., y_ = v s (meaning that x_ = 0) occurs. This happens at time t = slip := 2=!s + 2=!s . Rajchenbach (1990) and others estimate that slip

:= 2=!s and emphasize that it is independent on the velocity. This, however, is 31

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not perfectly true, slip depends on vs through . The slip duration is plotted as a function the driving velocity in Fig. 2.4 a. We have also plotted the velocity dependence of the time lapse between successive slip events (Fig. 2.4 b). At low velocities it scales with 1=vs, but at high velocities it becomes approximately constant. The presented curve is in good qualitative agreement with experimental data of Nasuno et al. (1998). Turning back to the real position x(t) of the block (2.6) reads:

x(t) = A cos(!st ) + vs t

Ff : ks

(2.9)

The general case of the slip dynamics is plotted by dashed lines in Fig. 2.3. The slip is no longer a half circle in x_ (y ), it is more than half period of oscillation. This simple model reveals, that the basic ingredients of stick-slip processes are the continuous driving, and that initiating relaxation is “harder” than maintaining it. These conditions are satisfied for many systems in nature (e.g. for all the aforementioned examples), explaining why stick-slip fluctuations are so widespread.

2.1.3 Transition to Continuous Motion A general feature of stick-slip motion is that if the driving velocity is increased, then a transition to continuous flow occurs. The squeaking sound of the door disappears if it is opened quickly; if sufficient electric current is provided, than a continuous discharge can be maintained; and also, last minute panic eliminates fluctuations in proceeding in writing a thesis. The simple explanation for this transition is that at intense driving the time lapse between successive slip events becomes smaller than the duration of a single slip, thus the slip events merge (Rajchenbach, 1990). This is a good qualitative picture, but we have to handle the question carefully, since as Fig. 2.4 b indicates, the time lapse between slip events is constant for high velocities. On the other hand, if starting from the continuous phase the velocity is decreased, then a back-transition to stick-slip motion occurs. In general the two transition points do not coincide, the transition is often hysteretic. Our minimal model described by equation 2.3 is unable to account for this transition, both the stick-slip phase and the continuous phase are stable for all velocities. In order to account for the observed transitions we have to introduce a more sophisticated friction law. Allowing velocity dependence and explicit time dependence in the friction law we write a generalized version of (2.3):

M x = ks (vs t x) Ff (x; _ t); 32

(2.10)

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a

Fmax F0

x' x

y

b

y'+vs (=x')

c

d

stick

stick

0 ystart

stick

0

t

t

ystart

0

y

-ystart

Figure 2.5: Behaviour of the stick-slip model if friction is linearly increasing with velocity. (a) The frictional force as a function of velocity. (b) Time evolution of the position x(t) of the block. Solid line corresponds to v s 0. Below a critical velocity stick-slip motion is stable (dotted line), at higher velocities the system approaches the steady state (dashed line). The straight lines show the steady state position x S:S: . (c) y (t) = x(t) xS:S: for the same velocities. (d) The velocity of the block as a function of its position (x_ (y )) calculated from the same data. which is now a general model for stick-slip processes. The properties of a given system

_ t). The explicit time dependence is necessary to describe irregucan be specified in Ff (x; lar stick-slip processes and systems, where F f (x_ ) is multivalued (e.g. shows hysteresis). In order to account for the transitions mentioned above, it is enough to use a form Ff (x_ ). Let us first consider a system, where the friction force increases linearly with velocity, Ff (x_ ) = F0 + x_ , where F0 and are constants, > 0. Then performing the same steps leads to:

M y =

if y_ = vs and 0 ks y y_ otherwise,

y < yslip

(2.11)

This is the equation of a damped oscillator. The resulting dynamics is shown in Fig. 2.5. If the velocity is small, then stick-slip motion is stable. However, if we reach a critical velocity, then due to the damping y_ cannot reach v s again, thus the block does not stop,

and the system turns into continuous motion. The critical velocity v c1 is thus strongly

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Fmax F0

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stick

b

y

y'+vs (=x')

c

stick

stick

stick

stick

0

d

0 ystart

t

t

ystart

0

y

-ystart

Figure 2.6: Behaviour of the stick-slip model if friction is a linearly decreasing function of velocity. (a) The frictional force as a function of velocity. (b) Time evolution of the position x(t) of the block. Solid line corresponds to v s 0. Stick-slip motion is stable (dashed line), continuous motion is unstable (dotted line) for all velocities. The straight lines show the (unstable) steady state position x S:S: . (c) y (t) = x(t) xS:S: for the same velocities. (d) The velocity of the block as a function of its position (x_ (y )) calculated from the same data. dependent on the damping coefficient , i.e. the slope of F f (vs ). The steady motion is stable for all velocities in this model. Performing a linear stability analysis of the steady state solution of (2.10) x S:S: =

Ff (vs )=ks + vs t reveals that it becomes unstable, when @Ff < 0: @vs

(2.12)

Indeed, if we modify our previous model so that < 0, then we obtain amplified oscillations (see Fig. 2.6). In such a model continuous flow is unstable for all velocities: infinitesimal perturbations are amplified until the block sticks. Thus in order to be consistent with experimental observations our friction law has to have a minimum as a function of velocity (e.g. at x_ = v ). If then the system is in continuous flow state and the velocity is decreased, then the instability happens at v2 = v . In the other direction the transition doesn’t necessarily happen at v , the critical

velocity v1 depends on the “steepness” of the Ff (v ) curve. If Ff (v ) is a quickly increasing 34

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function, then v1 v = v2 , if Ff (v ) is nearly constant, then v1 may be substantially larger than v = v2 — he observe a hysteresis in the transition. There has been much debate about the origin of stick-slip motion and the hysteretic stick-slip—continuous transition in various experiments and models (Jaeger et al., 1989; Rajchenbach, 1990; Thompson and Grest, 1991; Benza et al., 1993; Linz et al., 1999; Rajchenbach, 2000; Lubert and de Ryck, 2001). We hope that we have clarified the situation. For the appearance of stick-slip motion the necessary condition is that friction force at vs = 0 should be larger than at small velocities (either because of a discontinuity, or due to F 0 (0) = dF=dvs (0) < 0). To account for the transition to continuous flow a

velocity-dependent friction law has to be considered, the transition occurs if F 0 > 0 and the velocity is higher than a critical value determined by the actual form of F f (vs ). The origin of the back-transition is the existence of a finite optimal velocity, where the friction force is minimal ( and below which F 0 < 0 ). Finally we note that a velocity-dependent frictional force cannot account for all the observed phenomena (e.g. Nasuno et al. (1998) observed hysteresis in Ff (v )), in such cases further variables (e.g. dilation) have to be introduced. But the coupling of the frictional force to the velocity remains essential even if it happens indirectly through a third quantity.

2.1.4 Stick-Slip Processes in Granular Materials There are many examples of stick-slip behaviour in granular materials. In the first type of experiments stick-slip sliding between a granular medium and a solid body is considered. Perhaps the simplest, but thoroughly investigated and thus extremely revealing case is when a solid block is pushed on top of a granular layer (Nasuno et al., 1997; Nasuno et al., 1998; Géminard et al., 1999). In a different setup a granular sample is pushed by a spring inside a hollow cylinder (Ovarlez et al., 2001). In the second class of experiments the granular sample is forced to deform slowly, and intermittent motion is caused by successive discrete reorganization events inside the material. Experiments in elastic annular shear cells revealed both regular (Cain et al., 2001) and irregular (Miller et al., 1996; Dalton and Corcoran, 2001) stick-slip behaviour. The force required by a bidimensional piling to be compressed uniaxially was also reported to exhibit stick-slip fluctuations in the case of elastic coupling. If the coupling was stiff enough, continuous deformation with intermittent stick events occurred (Ngadi and Rajchenbach, 1998). In Chapter 3 we describe in detail our experiment on the fluctuation of the granular drag force, which is also governed by stick-slip internal friction. 35

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Finally, there is a third type of experiments, in which the driving consists of increasing the inclination angle of the material surface, and slip events are surface avalanches. The surface angle can be increased by either adding grains on top of a pile (Held et al., 1990; Liu et al., 1991; Rosendahl et al., 1994; Frette et al., 1996; Lemieux and Durian, 2000) or by tilting the material, which can be placed either into a rectangular container (Bretz et al., 1992; Evesque, 1993; Boltenhagen, 1999; Aguirre et al., 2000) or into a rotating drum (Jaeger et al., 1989; Nagel, 1992; Rajchenbach, 1990; Evesque, 1991; Dury et al., 1998). Though avalanches in a rotating drum usually are not thought about as stick-slip processes, that they exhibit all the important features of stick-slip systems, and they can be described by the stick-slip model (eqn. 2.10). All the experiments described above were performed in dry, cohesionless samples. However, if there are cohesive forces among the grains they qualitatively modify the behaviour of the material. Cohesion may arise either form Van der Waals interactions (Valverde et al., 2000), if the grain size is very small (in the m range), or from capillary forces, if some interstitial liquid is present. Many of our own experiments investigates the effect of wetness on the static (Chapter 4) and dynamical (Chapter 5) behaviour of granular materials. For this reason in the following we devote special attention to earlier studies on the possible presence of some liquid and its consequences. In the following sections we analyse granular stick-slip processes in more detail. We study the properties of the 3 major events separately: the stability of the sticking phase (Section 2.2), the dynamics of the slip process (Section 2.3), and the properties of the continuous motion occurring at intense driving (Section 2.4).

2.2 The Sticking Phase — Stability of Jammed States 2.2.1 Jamming We first consider the sticking phase, when the material is at rest under increasing load. As we have seen in Section 1.1.1, the resting state of granular materials is far from being trivial. When the resting state is created, the grains keep moving until they are organized into a structure, when the displacement of every grain is impeded by its neighbours — this process is called jamming (Cates et al., 1998; Bouchaud et al., 1998; Cates and Wittmer, 1999; D’Anna and Gremaud, 2001; Trappe et al., 2001). The particularity of a jammed state is that the most important factor of stability is the geometrical arrangement of the grains. The structure of the material is self-adaptive, the grains reorganize until they 36

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reach a state, in which the network of contact points is suitable to support the applied load. In practice this means that the force chains align themselves in the direction of the acting forces. A consequence of this self organization is that the material can hold only specific types of loading, if we apply forces that are incompatible with the current geometry of the grains (e.g. perpendicular to the force chains), then the grains will have to reorganize. It is an intriguing question, how the extremely inhomogeneous, frustrated structure of the material reacts to additional external forces, to what extent it can resist the higher forces by elastic deformation of the particles and the contact areas, and what is the threshold, when one or more of the grains become unstable and a reorganization occurs. Finding a general answer to that question is very difficult. The stability of the packing is strongly affected by the shape and size distribution of the particles (Nasuno et al., 1998; Nakagawa and Miyata, 2001), the packing fraction (Evesque, 1993), finite size effects (Dury et al., 1998; Boltenhagen, 1999; Aguirre et al., 2000), and construction history (Vanel et al., 1999), including the time elapsed since the construction of the packing (Losert et al., 2000). In the following we present only the simplest arguments on granular stability, which give a generally applicable description, but are unable to account for the above mentioned factors. Then we analyse a single one of these factors: the strengthening over time.

2.2.2 Stability of Dry and Wet Granular Materials The simplest and most widespread theory of the stability of granular materials is the Mohr-Coulomb analysis (see e.g. Nedderman, 1992 or any other soil mechanics textbook). It describes the granular sample as a continuum, the internal forces are characterized by a stress tensor, its eigenvalues are the principal stresses. Let us consider a small element of 2D granular material with the principal stresses 1 > 2 . Let us choose an

arbitrary straight line through this small element with an angle with respect to the direction of the eigenvector corresponding to 1 . It is easy to show (simple trigonometry),

that the normal and tangential stresses ( and respectively) on that line are:

= + cos(2) = sin(2);

37

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b

τ

stable unstable τ*

φ c

2α σ2

σ*

σ1

σ

Figure 2.7: Illustration of stability arguments of cohesive materials. (a) Mohr-Coulomb analysis. The failure occurs in the bulk of the material along a plane. (b) The arrangement considered by Hornbaker et al. (1997) to calculate the stability of a grain on the surface. where = (1 + 2 )=2 and = (1 2 )=2. Eqn. 2.13 means, that the possible stress values along for various lines lie on a circle (centered at ( ; 0), with a radius ) in the

(; ) plane, this is called the Mohr’s circle (see Fig. 2.7 a).

The Mohr-Coulomb stability criterion is analogous to the well known Coulomb friction law for two sliding surfaces. It postulates that along every line the stresses must satisfy

j j tan + c;

(2.14)

where is the angle of internal friction, c is the cohesion of the material. This equation defines a pair of straight lines in the (; ) plane, called the Mohr-Coulomb yield envelope. The material is stable, if Mohr’s circle is inside the yield envelope, failure occurs, when the circle touches the envelope. This criterion is very simple, it describes the material with only two variables, and c, still it gives a reasonable result in many situations. When applied for a sample with an inclined free surface, (2.14) allows us to determine

max , the maximum angle of stability. For a cohesionless material (c = 0) it predicts that max = , for a cohesive material max is larger. Eqn. 2.14 can also be used to predict the yield surface in an unstable sample. If c is large, it generally predicts that failure occurs along a line deep inside the sample, which indeed is the typical failure mechanism for cohesive materials (Valverde et al., 2000). In contrast, if dry samples are tilted so that the surface angle exceeds max , then the failure occurs on the surface: one of the grains starts rolling, and then it dislodges others. It is an intriguing question how the behaviour of the 38

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material changes if the liquid content of a sample is gradually increased from zero (Nase et al., 2001). Hornbaker et al. (1997) investigated samples consisting of glass spheres that contained an extremely small amount of oil (the ratio of liquid volume to the void volume among the grains was 10 4 ). For such small liquid contents one can observe no change in the motion of the grains, but there is a pronounced increase in the maximum angle of stability, max . They proposed an alternative description of the effect of cohesive forces based on the stability analysis of the grains on the surface, as shown in Fig. 2.7 (Hornbaker et al., 1997; Albert et al., 1997). They emphasize that when the capillary forces are calculated, the asperities on grains must be taken into account. Their analysis is only relevant for spherical grains and extremely low liquid contents. The Mohr-Coulomb analysis can also be applied to describe the mixture of glass balls and oil starting from the microscopic level of the grains. The task consists of finding a relation between the capillary forces due to liquid bridges between neighbouring grains, and the parameter c in (2.14). This link was established by Halsey et al. (1998, see also Mason et al., 1999). They calculate that c can be related to interparticle forces f A

as follows: c = fA z=(4R2 ), where z is the coordination number, is the packing fraction, R is the radius of the beads. By considering how the wetting fluid occupies the microscopically rough surfaces between the two spherical grains, they also predicted the value of fA as a function of the liquid volume per contact. They proposed 3 distinct regimes: fA (V ) is increasing in the asperity and roughness regimes according to different functional forms, and then it saturates in the spherical regime.

2.2.3 Aging As we emphasized earlier, there are several effect influencing the strength of a jammed state, which cannot be described by the simple models mentioned earlier (unless incorporated explicitly). We choose only one of these effects to investigate in more detail: the dependence of the stability of the material on the time for which it has been at rest. Several experiments reported, that the stability of the material is increasing proportionally to log( ). There are a number of mechanisms, that can explain such a logarithmic strengthening (see Losert et al., 2000 for a review), we briefly summarize the most important possibilities. Firstly, experiments on solid-on-solid friction have also revealed a logarithmic increase in the frictional force (Hähner and Spencer, 1998), which is related to a slow 39

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#

Figure 2.8: Avalanches triggered on an inclined plane. (a) Time evolution of a triangular avalanche. (b) Time evolution of an uphill avalanche. (c) shape of the avalanche front. (The figures are from Daerr, 1999.) change in the number and strength of the microcontacts that support the applied stress. Thus the friction coefficient between the grains may increase in time, leading to the observed strengthening. The second possibility is related to small reorganizations of the grains, as described by the concept of jamming (Sec. 2.2.1). According to this scenario the grains rearrange so that the direction of the grain-grain contacts is aligned in the direction of the applied force, and the increase in strength is related to the resulting stronger force-chain structure. A special case of this mechanism is when the rearrangement leads to compaction, but an increase in strength is possible without volume changes too. The third group of mechanisms is related to the presence and motion of an interstitial liquid, in particular the gradual development of liquid bridges among the grains. This mechanism accounts for the observed increase in strengthening at high humidities (Fraysse et al., 1997; Bocquet et al., 1998). The relative importance of these and possibly other effects may vary from one experiment to another.

2.3 The Slip Process — Internal and Surface Avalanches In the original stick-slip model the slip event is rather simple: the block is moving forward, while the spring is being relaxed. In contrast, in granular materials the slip event involves the motion a lot of grains. The reorganization is often initiated by the displacement 40

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of a single grain. This however initiates the motion of the neighbouring grains which then dislocate others, and the failure propagates and is amplified like an avalanche. If the failure occurs on an inclined surface, then indeed ordinary avalanches are formed, but propagating reorganizations inside the materials can also be termed “internal avalanches” (Manna and Khakhar, 1998).

2.3.1 Avalanche Size Distribution It is plausible that the size of avalanches is not necessarily always the same, the stick-slip motion may be highly irregular. Indeed, models inspired by the notion of self-organized criticality predict broad, power-law distributions of avalanche sizes. However, most experiments on granular surface avalanches revealed no self-organized critical behaviour, but indicated that the system can be better described by a regular stick-slip model with some fluctuations (Jaeger et al., 1989; Rajchenbach, 1990; Liu et al., 1991; Evesque, 1991). There exist a characteristic angle max (maximum angle of stability) where the avalanche is initiated, and also a characteristic stopping angle r , the repose angle. The distributions of the initial and final angles of avalanches are close to Gaussians centered at max and r . Thus the size of avalanches also follows a Gaussian distribution, though fluctuations may be large: the standard deviation of the distribution is often close to the mean value. There are however some systems which do exhibit avalanches with power-law size distributions. Held et al. (1990) reported SOC for very small piles. This could be explained by the fact, that in their experiment even the addition of a single grain increased the average angle by more than max r (Liu et al., 1991). Power-law distribution was also found for the small avalanche events between the large (max ! r ) avalanches (Bretz et al., 1992; Rosendahl et al., 1994) and for collapse events in a watered granular mixture (Somfai et al., 1994). Then Frette et al. (1996) revealed that the special shape of rice grains allow them to exhibit real SOC behaviour, which initiated a series of experimental (Christensen et al., 1996; Malthe-Sørenssen et al., 1999; Jia et al., 2000) and theoretical (Amaral and Lauritsen, 1996; Amaral and Lauritsen, 1997; Pastor-Satorras, 1997) work on the physics of rice-piles. Internal avalanches in sheared or deformed layers can be either regular (Cain et al., 2001), irregular (Miller et al., 1996) or critical (Dalton and Corcoran, 2001).

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(a)

a

c

(b)

b (c)

Figure 2.9: Comparison of different fronts. (a) The half flat hourglass experiment (Douady et al., 2001). (b) Superposition of successive surface profiles for the upper and lower part of the hourglass experiment. Bottom: the sand accumulates and then an avalanche starts with a sharp (“start down”) front propagating to the right. Later a “stop up” front propagates backwards. Top: The upper layer empties and the starting front propagates upwards (to the right), followed by a stopping front propagating down (to the left). (c) The fronts observed by Rajchenbach (2002). The front velocity is faster than the material flow, so grains do not accumulate at the front.

2.3.2 Avalanche Dynamics The detailed dynamics of the slip events is intriguing for all granular stick-slip systems (Nasuno et al., 1998; Lubert and de Ryck, 2001; Cain et al., 2001). However, avalanches on a free surface are of particular interest. A detailed experimental study of avalanches in dry materials was performed by Daerr and Douady (1999b). He investigated avalanches triggered in granular layers on a rough inclined surface. The initial inclination angle of the layer was in the metastable region between the maximum angle of stability max and the angle of repose r (both characteristic angles depends strongly on the layer width). In this region a pointlike perturbation initiated propagating fronts in the layer along which the particles were destabilized. Depending on the width of the layer and the inclination angle he found two different

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types of avalanches. At a suitable angle perturbing a thin layer results in an avalanche propagating downhill and also laterally owing to collisions between neighbouring grains, and leaving triangular tracks behind (Fig. 2.8(a)). In contrast, perturbation of a thicker layer results in an avalanche front that also propagates upwards, the grains located uphill progressively tumbling down because of loss of support (Fig. 2.8(b)). The amplitude and velocity of the avalanche saturates even though it gains material continuously while moving down. Both the head front and the rear of the avalanche moves with a constant velocity. The head front is steep, a pronounced height maximum is observed close to it (Fig. 2.8(c)). Behind it both the height and the velocity of the moving material is decreasing gradually, while their ratio is almost constant. In his measurements the velocity of the particles is about the same as the group velocity of the front. In a different setup (Fig. 2.9 a) Daerr identified 4 different types of fronts related to the avalanches depending on if the motion is started or stopped at the front and if the front propagates downhill or uphill (Fig. 2.9 b). The triangular avalanches thus are initiated by a “start down” front, which is also present in the uphill avalanches, but there a “start up” front occurs too. In a system with an open lower boundary the avalanche typically terminates by a “stop down” front: when the bulk of the avalanche has left the grains left behind settle and stop. In case of a closed boundary the incoming grains accumulating at the bottom wall increase the local height there, which slows down the grains coming behind. This is a “stop up” front, identical to the “kink” seen in earlier experiments (Makse et al., 1997; Gray and Tai, 1998; Cizeau et al., 1999). Rajchenbach (1998,2000,2002) performed experiments, where the avalanche was triggered in a different way: the surface was tilted until its angle became larger than

max (Fig. 2.9 c) . His results indicate that the initiation of the avalanche is somewhat similar to the uphill avalanches: first a single grain is destabilized, and this perturbation

propagates both upwards and downwards with a constant velocity 40 cm/s. However, in contrast with the observations of Daerr, in this case the front propagation is quicker by a factor of two than the material flow, and thus no material is accumulated at the front. The reason for the difference is probably that in this case the grains are much closer to instability, and blocks of material are destabilized together. When the front reaches an unstable block, the shock on the first grain of the block is quickly transmitted to the other particles of the block destabilizing all of them and the avalanche front suddenly jumps forward. Within the unstable blocks the propagation of the front is due to collisions, i.e. only momentum transfer not material flow.

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These intriguing experiments stimulated theoretical research and much of the experimental findings has been successfully interpreted (Aronson and Tsirming, 2001).

2.3.3 Dilation Since Bagnold (1966) the bistable and hysteretic behaviour of granular materials is often attributed to dilatancy effects. Indeed, sensitive measurements revealed that slip events in granular materials are always accompanied by a small dilation, though the exact time dependence of the process may vary from one experiment to another. In experiments with a slider dilation occurred right before the slip events (Nasuno et al., 1998). In contrast, shear experiments (Cain et al., 2001; Lubert and de Ryck, 2001) revealed a gradual increase of the material volume all along the stick period, and the material was actually compacted during the slip. The fronts observed by Rajchenbach also correspond to a dilation of the system. The dilation of the material during the slip is not only an interesting side effect, its presence plays a crucial role in other observed phenomena. For example the detailed dynamics of the slider experiment (Nasuno et al., 1998), namely the hysteretic dependence of the frictional force on velocity could only be explained by introducing an extra variable to describe dilation (Lacombe et al., 1999; Géminard et al., 1999).

2.4 Continuous Motion — Granular Flows If the rate of driving is increased above a threshold a transition to continuous motion can be observed in practically all granular stick-slip systems. When a granular system is subjected to high shear, it becomes fluidized and will move continuously. If the angular velocity of a rotating drum is high enough, then the intermittent avalanches merge and we observe a continuous flow. Similarly, steady flow can be maintained if a sufficient material flux is poured continuously from a point source. The transition can be either sharp (Nasuno et al., 1998; Lemieux and Durian, 2000), gradual (Cain et al., 2001; Lubert and de Ryck, 2001) or hysteretic (Rajchenbach, 1990; Rajchenbach, 2000) depending on the details of the system. We showed earlier, that a necessary condition for a hysteretic transition between stickslip and continuous motion is that the friction force should have a minimum as a function of velocity. Jaeger et al. (1990) proposed a model that yielded such a friction law. They suggested two different origins for the dissipation. The first one is due to the collisions 44

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Velocity

Figure 2.10: The velocity-dependence of the frictional force in the model of Jaeger et al. (1989). which leads to a shear force fcol / (rv )2 (Bagnold, 1954). This term is dominant at high velocities. The second term takes into account that at low velocities the grains fall into the corrugations of the layer below. The loss of potential energy in such cases leads

to shear force fcol / 1=(1 + (rv )2 ). The sum of the two terms lead to a total friction force that has a minimum at a finite velocity (see Fig. 2.10):

Ff (rv ) =

a + b (rv )2 : 1 + (rv )2

(2.15)

Several experimental studies were performed recently on continuous surface flows, where the flow occurs due to gravity on the surface of a fixed bed. One of the moth interesting questions concerned the depth dependence of the velocity of the grains. Most studies reported that the velocity profile is linear, and its gradient is independent on the flow rate (Rajchenbach, 1998; Bonamy et al., 2001; Orpe and Khakhar, 2001). Recently Komatsu et al. (2001) demonstrated the existence of an exponential tail below the linear region. Different velocity profiles were reported in rotating drum experiments with only few large grains (Yamane et al., 1998) and in thin flowing layers (Lemieux and Durian, 2000; Pouliquen, 1999). Several heuristic arguments have been proposed to explain the linear velocity profile (Jaeger et al., 1989; Rajchenbach, 1998). Andreotti and Douady (2001) reproduced both the linear and the exponential region by assuming non-local dissipation due to shocks and trapping of beads by irregularities of the underneath layer.

45

Part II NEW RESULTS

46

Chapter 3 Stick-Slip Fluctuations in Granular Drag 3.1 Introduction Jamming in granular materials has originally been discussed in the context of the gravitational stress induced by the weight of the grains (Cates et al., 1998; Bouchaud et al., 1998; Cates and Wittmer, 1999, see also Sec. 2.2.1). However it can result from any compressive stress. For example, a solid object being pulled slowly through a granular medium is resisted by local jamming, and can only advance with large scale reorganizations of the grains1 . The granular drag force originates in the force needed to induce such reorganizations, and thus exhibits strong fluctuations which qualitatively distinguish it from the analogous drag in fluids (Albert et al., 1999). In this chapter we investigate the dynamic evolution of jamming in granular media through these fluctuations in the granular drag force. The successive collapse and formation of jammed states give a stick-slip character to the force with a power spectrum proportional to 1=f 2 . We find that the stick-slip process does not depend on the contact surface between the grains and the dragged object, demonstrating that the forces arising from jamming of the grains and subsequent bulk grain reorganization dominate the drag process while the frictional forces at the surface have little effect. While the fluctuations are remarkably periodic for small depths, they undergo a transition to ”stepped” motion at large depths. These results point to the importance of the long-range nature of the force 1

It is possible , however, for an object to move through a vibrated granular medium without large scale reorganizations: (Duran et al., 1993)

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Figure 3.1: (a) A sketch of the apparatus used for measuring the drag force. chains to both the dynamics of granular media and the strength of the granular jammed state. This work was initiated at the Notre Dame University, the preparation of the experimental setup and measurements of time averaged drag force are due to Albert et al. (1999). During my visit of 6 months in the USA I took over the experiment, made improvements on the apparatus, performed preliminary measurements, and initialized the analysis of fluctuations. Later I took serious part in the analysis and the preparation of the published paper. However, the bulk of the measurements presented here were performed by István Albert. For this reason I restrict myself to a brief description of our findings here, more details can be found in our publications (Albert, Tegzes, Kahng, Albert, Sample, Pfeifer, Barabasi, Vicsek, and Schiffer, 2000; Albert, Tegzes, Albert, Sample, Barabási, Vicsek, Kahng, and Schiffer, 2001).

3.2 The Experimental Setup The experimental apparatus, shown in Figure 3.1, consists of a vertical steel cylinder of diameter dc inserted to a depth H in a cylindrical container filled with glass spheres

(obtained from Jaygo Inc., Union, NJ). A comb made of three 5 mm diameter steel rods separated by 20 mm gaps is also inserted at 150 mm depth opposite the vertical cylinder

with the role of randomizing the medium. This comb has the additional benefit of stabilizing the packing fraction during the experimental runs. The container rotates with constant angular speed while the vertical cylinder is attached to an arm that may rotate freely around the rotational axis of the container. Opposing the cylinder we have a fixed 48

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H = 60mm

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H = 80mm

7 6 H = 100mm 18 16

H = 160mm 16

17

18

19

20

21

Distance (mm) Figure 3.2: The characteristic fluctuations in the drag force at 4 different values of H for dc = 10 mm. Note the transition from purely periodic fluctuations H 60 to stepped fluctuations with increasing depth H 100. precision force cell which measures the force F (t) acting on the cylinder as a function of time. We also incorporate a spring of known spring constant k between the cylinder

and the force cell with k varying between 5 and 100 N/cm. The purpose of the spring is twofold; it allows the force to slowly build up, and with a suitable choice of k it dominates the elastic response of the cylinder and all other parts of the apparatus so that the nonlinear deformations will not significantly affect the results. We vary the speed (v ) from 0:04 to 1:4 mm/s, the depth of insertion H from 20 to 190 mm, and the cylinder

diameter (dc ) from 8 to 24 mm, studying glass ( = 2:5 g/cm 3 ) spheres of diameter (dg ) 0:3, 0:5, 0:7, 0:9, 1:1, 1:4, 1:6, 3:0, and 5:0 mm. (Unless otherwise noted, data are shown

for dg = 0:9 mm, k = 25 N/cm and v = 0:2 mm/s.) The force is sampled at 150 Hz and the response times of the force cell and the amplifier are less than 0:2 ms. Note that these experiments are conducted in dense static granular media as opposed to drag in dilute or fluidized media which have been studied both theoretically (Buchholtz and Poschel, 1998) and experimentally (Zik et al., 1992).

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3.3 Stick-slip Internal Friction Consistent with earlier results (Albert et al., 1999), we find that the average drag force on

the cylinder is independent of v and k , and is given by F = gd c H 2 , where characterizes the grain properties (surface friction, packing fraction, etc.), is the density of the

glass beads, and g is gravitational acceleration. As shown in Fig. 3.2, however, F (t) is not constant, but has large stick-slip fluctuations consisting of linear rises associated with a compression of the spring and sharp drops associated with the collapse of the jammed grains opposing the motion. The linear rises in F (t) correspond to the development of an increasingly compressed jammed state of the grains opposing the motion. We find that,

independent of the depth, the slopes of the rises are given by v1 dF=dt = k for all springs with k < 100 N/cm, confirming that the spring dominates the elasticity of the apparatus.

This result also implies that, during the rises, the jammed grains opposing the cylinder’s motion do not move relative to each other or the cylinder. The power spectra, P (f ) (the

squared amplitudes of the Fourier components of F (t)), are independent of both the elasticity of the apparatus and the rate of motion, so that the scaled spectra kvP (f ) vs. f=kv collapse in the low frequency regime, f < 10 Hz (Miller et al., 1996). This indicates that the fluctuations reflect intrinsic properties of the development and collapse of the jammed state rather than details of the measurement process. The power spectra also ex-

hibit a distinct power law, P (f ) / f 2 , over as much as two decades in frequency (Fig. 3.3) a phenomenon which has been reported in other stick-slip processes and is intrinsic to random sawtooth signals (Reiter et al., 1994; Demirel and Granick, 1996; Feder and Feder, 1991). The observed behaviour is identical to other stick-slip processes described in Chapter 2. During each fluctuation the force first rises to a local maximum value (F max ), and then drops sharply (by an amount F ), corresponding to a collapse of the jammed state. The force from the cylinder propagates through the medium via chains of highly strained grains, and a collapse occurs when the local interparticle forces somewhere along one of the chains exceed a local threshold. The corresponding grains then slip relative to each other, which in turn nucleates an avalanche of grain reorganization to relieve the strain. This allows the cylinder to advance relative to the granular reference frame, with a corresponding decompression of the spring and a drop in the measured force F (t). (We see no evidence of precursors (Nasuno et al., 1998) to the slips within our time resolution of 0:2 ms.)

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Figure 3.3: The power spectrum of the fluctuations in the drag force taken for dg = 1:1 mm and dc = 16 mm at depth H = 60 mm with v = 0:05 mm/s and k = 5 N/cm to increase the dynamic range. Note that in both the periodic and the stepped regimes the spectrum has a long f 2 regime as shown in the upper inset (shifted vertically for clarity) (dc = 16 mm). The lower inset (dg = 1:1 mm, dc = 19 mm) shows that the power spectrum does not change when a half-cylinder is substituted for a full cylinder. The interparticle forces within the force chains are largest at the cylinder’s surface, where the chains originate, and their magnitude decreases as we move away from the cylinder and the force chains bifurcate. Consequently, one might expect that the reorganization is nucleated among grains in contact with the cylinder, but we find no change either in F or in the fluctuations when we vary the coefficient of friction between the grains and the cylinder by a factor of 2:5 (substituting a teflon-coated cylinder for the usual steel cylinder). As demonstrated in the inset to Fig. 3.3, the power spectra are also unchanged even by substituting a half-cylinder (i.e. a cylinder bisected along a plane through its axis and oriented so that plane is normal to the grain flow) for a full cylinder of the same size, indicating that the geometric factors do not play a significant role either. These results indicate that the fluctuations are not determined by the interface between the dragged object and the medium, but rather that the failure of the jammed state is nu51

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120 80 40 80

120 160 200

∆ F (N)

40

0.6

F

10

1

Slope = 2.0 0.1

10

0.0 20

40

60

80

100

100 120 140 160 180 200

Depth (mm) Figure 3.4: The transition from periodic to stepped fluctuations as shown through the magnitude of the average drop F , for two different container sizes: Æ - large container (D =25mm), N - small container (D = 10 mm). The upper inset shows the relative magnitude of the standard deviation n = F =F . The transition occurs at smaller Hc in the smaller container. (dc = 10 mm) The lower inset shows the depth dependence of the average drag force for dg = 1:1 mm and dc = 10 mm, there is no change in slope at Hc. The solid line has slope 2. cleated within the bulk of the medium. In this respect, the fluctuations are rather different from either ordinary frictional stick-slip processes which originate at a planar interface between moving objects, or the motion of a frictional plate on top of a granular medium (Nasuno et al., 1997; Nasuno et al., 1998; Géminard et al., 1999).

3.4 Transition to “Stepped” Fluctuations A striking feature of the data is that the fluctuations change character with depth. For H < Hc 80 mm the fluctuations are quite periodic, i.e. F (t) increases continuously

to a nearly constant value of Fmax and then collapses with a nearly constant drop of F (Fig. 3.2). As the depth increases, however, we observe a change in F (t) to a ”stepped” 52

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signal: instead of a long linear increase followed by a roughly equal sudden drop, F (t) rises in small linear increments to increasing values of F max , followed by small drops

(in which F is on average smaller than the rises), until F max reaches a characteristic high value, at which point a large drop is observed. This transition from a periodic to a ”stepped” regime is best quantified in Fig. 3.4, where we plot the depth dependence of

F and the relative standard deviation of F , n = F =F . In the periodic regime, F rises due to the increase in F . As the large uniform rises of the periodic regime are broken up by the small intermediate drops, however, F shows a local minimum and F =F increases drastically, saturating for large depths. The transition is also observed in the power spectra as shown in the upper inset to Fig. 3.4. For low depths the power spectra display a distinct peak characteristic of periodic fluctuations, but these peaks are suppressed for H > character of F (t).

Hc in correlation with the changes in F =F and the qualitative

3.5 Discussion The transition from a periodic to a ”stepped” signal is rather unexpected, since it implies qualitative changes in the failure and reorganization process as H increases and the existence of a critical depth, Hc . An explanation for Hc could be provided by Janssen’s law (Brown and Richards, 1970) which states that the average pressure (which correlates directly with the local failure process) should become depth independent below some critical depth in containers with finite width. This should not occur in our container, however, which has a diameter of 25 cm, much larger than Hc . Furthermore we see no deviation in the behaviour of F (H ) from F / H 2 , which depends on the pressure increasing linearly with the depth (Fig. 3.4 lower inset). In order to account for the observed transition, we must inspect how the force chains originating at the surface of the cylinder nucleate the reorganizations. The motion of the cylinder attempting to advance is opposed by force chains that start at the cylinder’s surface and propagate on average in the direction of the cylinder’s motion. These force chains will terminate rather differently depending on the depth at which they originate, as shown schematically in Fig. 3.5. For small H , some force chains will terminate on the top surface of the granular sample and the stress can be relieved by a slight rise of the surface. Force chains originating at large depths, however, will all terminate at the container’s walls. Since the wall does not permit stress relaxation, the grains in

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Figure 3.5: (b) Schematic illustration of the force chains originating at the surface of an object dragged to the right in a finite container, where Hc corresponds to the depth at which the force chains all terminate at the walls of the container. Of course, this picture is highly simplified since the actual force chains bifurcate and follow non-linear paths. these force chains will be more rigidly held in place. According to this picture,

Hc

corresponds to the smallest depth for which all force chains terminate on the wall. When the cylinder applies stress on the medium, the force chains originating at small H (H
H c

impedes such microscopic relaxations. Thus a higher proportion of the total force applied by the cylinder will be supported by those force chains, enhancing the probability of a local slip event occurring at high depths. Such a slip event would not necessarily reorganize the grains at all depths (for example the grains closer to the surface may not be near the threshold of reorganization), thus the slip event might induce only a local reorganization and a small drop in F (t). The large drops in F (t) would occur when force chains at all depths are strained to the point where the local forces are close to the threshold for a slip event. This scenario also explains why F (H ) does not change at H c , since F is determined by the collective collapse of the jammed structure of the system. According to this picture, the transition is expected at smaller depths in smaller containers since the force chains would terminate on the walls sooner (see Fig. 3.5). Indeed, as we show in Fig. 3.4, the transition does occur at a depth approximately 20 mm smaller

when the measurements are performed in a container 2:5 times smaller (with diameter of 54

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100 mm). Furthermore, we fail to observe the periodic fluctuations in any grains with diameters 1:4 mm or larger, which is consistent with the suggested mechanism, since larger grains correspond to a smaller effective system size. It is interesting to compare our results with those of Miller et al., 1996 who studied fluctuations in the normal stress at the bottom of a sheared granular annulus. Those fluctuations were also independent of the rate of the motion, and demonstrated the longrange nature of the vertical force chains. In the present experiments, we confirm that force chains originating from a horizontal stress are also long range through our observation of the transition at depth Hc . Since for small grains (dg 1:1 mm) Hc has no measurable dependence on grain size, we find in agreement with Miller et al. that the nature of the force chains is not strongly dependent on grain size. Our observations also shed light on the implications of the long-range force chains for granular dynamics and the nature of jamming in granular materials. The crossover at H c suggests that drag fluctuations in an infinitely wide container would be periodic, but that the finite size of a real container destroys the periodicity. In other words, the finite size of a container relative to these chains reduces the strength of jammed granular states within the container. These results point to the need for a better understanding of the detailed dynamics of force chains — both how they form when stress is applied to a granular medium and how they disperse geometrically from a point source of stress — in order to gain an understanding of slow granular flows.

55

Chapter 4 Liquid-Induced Transitions in Granular Media The wide variety of novel dynamic and static physical behaviour presented in Section 1.1 was observed almost exclusively in dry granular media, and most of the related theoretical research was also restricted to dry samples. Dry grains interact primarily through two forces: elastic repulsion and friction. The presence of a thin layer of liquid on the grains, however, adds a third interaction to the problem - an attractive force, comparable in magnitude to the other two. The primary effect of this attractive force is that it effectively increases the stability of the material. In this chapter we investigate this strengthening of the jammed states due to the controlled addition of liquid, and identify fundamental changes in behaviour. The succeeding chapter describes, how these mechanisms are manifested in a stick-slip process.

4.1 Introduction The presence of an interstitial liquid adds a new dimension and complexity to the underlying physics of granular materials (Albert et al., 1997; Barabási et al., 1999; Halsey and Levine, 1998; Nedderman, 1992). Despite the technological ramifications (many industrial applications involve humid environments or liquid-coated grains), systematic experimental studies of the physical properties of wet granular materials have only been performed in recent years (Hornbaker et al., 1997; Bocquet et al., 1998; Fraysse et al., 1997; Alonso et al., 1998; Mason et al., 1999; Bocquet et al., 1998; Fraysse et al., 1999; Nase et al., 2001; Samadani and Kudrolli, 2000).

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A granular property which is strongly affected by the addition of liquid is the angle of repose (R ), the characteristic angle exhibited by a granular slope after an avalanche. Recent experiments (Hornbaker et al., 1997) demonstrated that even a nanometer-scale coating of liquid on millimeter-size grains can result in large changes in R . The experimental data could be fit by a theory based on the stability of grains on the top surface (Albert et al., 1997; Barabási et al., 1999), but an alternative theory linking the microscopic capillary forces to bulk stability arguments has also been proposed by Halsey and Levine (1998) (for a brief overview of the two theories see Sec. 2.2.2). An important difference between these two models is that the former predicts R to depend only on

the local surface properties, while the latter predicts that R should depend on the bulk properties of the grain pile as a whole — such as the size of the system in which R is being measured. We have performed detailed measurements of R as a function of liquid content and sample size to test these theories (Tegzes, Albert, Paskvan, Vicsek, and Schiffer, 1999), and we find that the physical behaviour of wet granular media is richer than anticipated by either approach. In particular, we observe three distinct characteristic regimes depending on the liquid content. We also find that the two aforementioned theories can each qualitatively describe the behaviour, but in different regimes. Our data shed light to how the increasing liquid content leads to a gradual change from the behaviour characterizing dry samples to the qualitatively different phenomena of wet materials.

4.2 The Experimental Setup Our experiments consist of a series of measurements of

R performed by the draining

crater method (Brown and Richards, 1970) which is depicted in Fig. 4.1. In this method a cylinder of diameter (dC ) is filled to some height, h0 (dC =2) tan(R ), with grains of a known quantity and packing fraction. A circular aperture is then opened in the bottom of the cylinder which allows the grains to drain from the cylinder — first forming an annular crater and then draining in flow down the surface of the crater until static stability is reached. We then determine R (the average slope of the stable crater surface) by weighing the grains which have drained. In order to compare the experiment to the two theories mentioned above, we made measurements of R using three different cylinders with dC = 10:3 cm, 15:6 cm, and

20:4 cm. Similarly to the drag force measurements, our granular medium consisted of 57

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Figure 4.1: A depiction of the draining crater method for measuring the angle of repose. (a) The initial filling of the container to a depth h0; (b) The first stage of draining through “bulk flow”; (c) The second stage of draining through “surface flow”; (d) The final stable crater with slope at the angle of repose (R ). spherical glass beads of average diameter 0:90 0:05 mm with surface roughness 1 m. The draining aperture was sufficiently large (diameter = 2:5 cm) that the medium drained in every case. (For smaller apertures, the adhesion of the grains can prevent draining (Hornbaker et al., 1997)). To minimize the effects of evaporation, we used vacuum pump oil as a wetting liquid with an uncertainty of < 3 % in the added liquid content. (The oil was obtained from Duniway Stockroom, its product id. was MPO-190,

its chemical formula: C2 H6 (CH2 )n where 20 n 40.) We cite the nominal average oil film thickness (Æliq ) as the average thickness of the liquid assuming that the spheres are perfectly smooth, uniform in size, and that the liquid is uniformly distributed on the grains. Note that our values of Æliq are thus overestimates of the actual layer thickness since we do not account for the surface roughness. The range of liquid content studied

was 0 Æliq 275 nm, which is equivalent to the liquid filling of interstitial volume between the grains. 1

0:27 % of the

We mixed the 5 kg granular samples for more than 30 minutes before measuring in order to guarantee that the liquid was well-distributed among the grains, and we found that the results did not depend on the amount of further mixing. Our data also did not depend on the delay time (typically less than 1 minute) between pouring the mixed sample into the cylinder and the beginning of the draining process, even when the delay was extended up to 10 minutes. The time period (usually less than 10 min) between the end of The conversion between Æ liq and %volume saturation is: V % = 300(Æ liq=R)=(1 ), where V % is the %volume saturation, R is the average radius of the grains, and is the packing fraction. 1

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45

ΘR (degrees)

dC = 20.4 cm 40 35 24

30 22

25 20 0

5 10 15 20 25

20 0

50

100

150

200

250

300

δliq (nm) Figure 4.2: The angle of repose as a function of the nominal average oil film thickness (Æliq ) with three different container diameters (, dC = 10:3 cm; M , dC = 15:6 cm; , dC = 20:4 cm). The two vertical arrows indicate the transitions between the granular, correlated, and viscoplastic regimes. The three dashed lines are fits to the bulk theory (Halsey and Levine, 1998) for the three different container sizes used. The inset shows an enlargement of the small Æliq regime. the mixing and the beginning of the experiment was also found not to affect the data, even for times up to 100 minutes. This indicates that the flow of the liquid to the contact points was unimportant to the measurements, although on a longer timescale (2 3 days) some aging was observable for the highest oil contents. The procedure for filling the cylinder was strictly controlled so that packing fraction in the samples was 0:63 0:01 (independent of Æliq ), although we did not find R to vary beyond the experimental uncertainty if the filling procedure was changed slightly.

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4.3 Three Regimes of Behaviour 4.3.1 The Repose Angle In Fig. 4.2 we show our raw data of R vs. Æliq which displays three regimes with different characteristic behaviour in each. At the lowest liquid contents, in what we will call the granular regime (Æliq < 20nm), R (Æliq ) is linear in Æliq and does not depend on the container size. For larger liquid contents, in what we will call the correlated regime

(20 Æliq 175nm), R (Æliq ) has a clear negative curvature and decreases for larger containers. At the largest liquid contents (Æ liq 175nm), which we call the viscoplastic2 regime, R (Æliq ) first decreases and then increases slightly with Æliq , but still depends on dC . The boundaries between the three regimes can be seen clearly in Fig. 4.3 a where we plot the numerical derivative of R , d(R )=d(Æliq ), which has a sharp peak at the

transition at Æliq

30nm and also a broader negative peak at Æliq 200nm.

4.3.2 The Draining Process The dynamics of the draining process also reflect the above three regimes and can be quantified by the time required for draining the cylinder, t drain , which is plotted in Fig. 4.3 b. As shown in Fig. 4.1, the draining process consists of two stages, “bulk flow”, which lasts for time tbulk , and “surface flow” which lasts for time tsurf such that tdrain = tbulk + tsurf . During bulk flow (Fig. 4.1 b) the aperture is covered by the grains, and the flow rate is limited by the aperture size and the pressure on the grains over the aperture. During surface flow (Fig. 4.1 c), the aperture is no longer covered by grains, and flow is along the exposed crater surface. We found by varying the initial mass of grains in the cylinder, that tbulk depends only weakly on Æliq (less than 10 % change over the full range

of Æliq ). Consequently the strong Æliq dependence of tdrain can be attributed to changes in tsurf .

In the granular regime the surface flow is homogeneous around the crater and appears to involve only the top few layers of grains as in the case of dry granular media (de Gennes, 1999; Jaeger et al., 1996; Wolf, 1996). Indeed, in this regime, t drain is dominated

by this relatively inefficient surface flow, and the decrease in tdrain with increasing Æliq possibly corresponds to the increasing thickness of the flow layer, which decreases t surf . In the correlated regime, the surface flow becomes strongly correlated in that clumps of 2

in (Tegzes et al., 1999) we used the term “plastic” for this regime. In fact the material is probably closer to a viscoplastic than to a pure plastic behaviour.

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many attached grains fall in each avalanche. Such avalanches often nucleate at some point on the surface and then the instability travels laterally around the crater. The final crater is typically highly anisotropic with the height at the cylinder wall varying by 5 15 mm, and the crater surface is rough with pronounced ridges and depressions. The large individual avalanches transport the grains more efficiently to the aperture, decreasing t surf

so that tdrain is determined primarily by the bulk flow rate (which is almost independent of Æliq ). In the viscoplastic regime, the surface flow is reminiscent of a viscous fluid in that the medium retains a smooth crater surface and the motion is coherent, draining at the same speed on all sides of the crater. Visual observation of draining in a transparent

container demonstrates, however, that flow in this regime occurs within a surface layer at the crater surface. When the draining ceases in this regime, the surface layer undergoes a slight elastic contraction analogous to the contraction of a dripping viscous fluid after a drop has fallen (see for example Dupont93). In this regime, the moving layer does not break loose from the surface like the avalanches of the correlated regime, but moves slowly along the surface. Thus the surface flow is apparently slowed by adhesive forces, and tsurf increases so that it once again determines tdrain .

4.3.3 Fluctuations

R was performed 30 times to obtain an average, we can also examine , the standard deviation of the measured values of R , which is plotted in Fig. 4.3 c. As can be seen in the figure, is small in the granular and viscoplastic Since each measurement of

regimes but much larger in the correlated regime. We attribute the differences between the three regimes to the dynamic nature of the surface flow processes. In the granular and viscoplastic regimes the surface remains smooth, and the flow of the grains is continuous throughout the draining process, and thus the system evolves adiabatically to its final state. By contrast, in the correlated regime the surface flow is dominated by individual avalanches, the character of which is determined by the particular arrangement of the grains at the start of a given draining process. The relatively large value of in the correlated region corresponds to the resulting roughness on the final surface of the crater.

4.4 Comparison to Theories While the experimental results clearly indicate the existence of three major regimes in wet granular media, we now must address the question of how to understand the regimes 61


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δliq (nm) Figure 4.3: Various measured properties of our system as a function of the nominal average oil film thickness, Æliq (, dC = 10:3cm; , dC = 15.6 cm; , dC = 20.4 cm). The approximate boundaries between the granular, correlated, and viscoplastic regimes are indicated by the shaded regions. (a) The numerical derivative, d( R )/d(Æliq ), based on the data in Fig. 4.2. (b) The total time required to drain the apparatus, t drain , as discussed in the text. (c) The standard deviation, , in the measured angle of repose. The much larger value in the correlated regime reflects the roughness of the surface. (d) The position of the FT-IR line associated with the antisymmetric C-H stretching mode in the oil on our grains. The increase at small Æliq corresponds to the increase in the liquid fraction of the film.

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in the context of the existing theories. The characteristics of R (Æliq ) in the granular and the correlated regimes are qualitatively similar to the predictions of the surface (Albert et al., 1997; Barabási et al., 1999) and the bulk (Halsey and Levine, 1998) theories respectively, although neither can fully describe the entire data set. In the granular regime, the observation of smooth surface flow, the linearity of R (Æliq ), and the independence

of R (Æliq ) on dC , are all consistent with the expectations of the surface theory. Some quantitative discrepancies persist, however, since we find R (Æliq = 0) 21Æ – about

10 % smaller than the theoretical prediction (Albert et al., 1997; Barabási et al., 1999). Furthermore, this theory only applies when R is determined by the behaviour of indi-

vidual grains on the surface, and thus cannot account for the behaviour in the correlated regime where the dynamics of correlated clumps dominate the surface flow. In the correlated regime, the existence of bulk avalanches, the curvature of R (Æliq ), and the decrease of R with dC are all consistent with the bulk theory within which one expects that R is determined by structural failure within the bulk of the material (Halsey and Levine, 1998). We attempted to fit the theory to the entire data set, and the best fits are shown by the lines in Fig. 4.2.3 While we find that it overestimates the changes in R due to the different dC (especially in the granular regime where we observe no dependence of R on dC ), the theory

does describe the basic qualitative features of R (Æliq ) in the correlated and viscoplastic regimes with the exception of the drop in R at the transition to the viscoplastic regime. The quantitative disagreement in these regimes may be due to the cylindrical experimental geometry, which is not incorporated into the theory. One other potential source of disagreement is that both theories predict the maximum angle of stability rather than the repose angle which we measure, but the stability of the final surfaces of the samples suggest that the differences between the two angles are not much beyond the uncertainties in the data. While the two theories can qualitatively describe the data in the three regimes, there is an outstanding question as to the origin of the transitions between the regimes. Although the nanometer-scale films are difficult to probe directly, we performed Fourier-transform infrared (FT-IR) spectroscopy4 on the oil to determine the structural nature of the films. 3

We fit the data to the theory of Halsey and Levine (1998) by treating as free parameters their roughness parameters (lR , c, and d) and the surface tension ( ). This fit reproduces most of the qualitative features of the behaviour in the correlated and viscoplastic regimes, although the necessary value of ( 0:09N/m) is almost three times larger than expected. The fit could not be improved by fixing the transitions between their three regimes to coincide with the three regimes we observe in the data. 4 FT-IR spectroscopy was performed with a Perkin-Elmer, Paragon 1000.

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We find that the frequency of the absorption line associated with the antisymmetric C-H stretching mode shifts upwards for Æ liq 30 nm and then saturates (see Fig. 4.3 d). Such a shift is consistent with the expected difference in the absorption line for molecules in solid and liquid states (Ulman, 1991), and these data suggest that the film consists of a localized wetting layer under a delocalized liquid layer which comprises a growing fraction of the total film volume as Æliq increases. We speculate that the delocalized liquid layer is covering an increasing fraction of the grain surface through continuous percolation as Æliq increases, and the increasing R (Æliq ) corresponds to the increasing coverage of the grain surfaces by this layer. Assuming that the intergrain attractive force is proportional to Æliq in this percolative regime, the surface theory predicts that R should increase linearly with Æliq which is consistent with our experimental results in the granular regime. The transition to the correlated regime then corresponds to a complete coating of the grain surface with a delocalized liquid film. In this case the increase in force due to the onset of capillarity (Fuji et al., 1998) would account for the peak in d( R )=d(Æliq ) and also the onset of the bulk correlated behaviour. We speculate that the transition to the viscoplastic regime, in particular the decrease in R and the smooth nature of the flow, corresponds to the onset of lubrication effects, something which is not incorporated in the existing theories. In summary, we observe three distinct regimes of coverage in the physical behaviour of wet granular media. This complex behaviour is not described by existing theories, and it suggests that the addition of liquid would have strong and non-trivial effects on other granular properties (e.g. size segregation etc.). Given the richness of the physics of wet granular media and their technological importance, we expect that the present results will provide a framework for further research on the topic.

64

Chapter 5 Avalanche Dynamics in Wet Granular Materials In the previous chapter we identified three fundamental regimes for the behaviour of wet granular materials as a function of the liquid content. Now we proceed and investigate how these regimes influence a spectacular stick-slip process: the dynamics of the free surface of a granular sample in a rotating drum. The rotating drum apparatus (see Fig. 5.1) is a cylindrical chamber partly filled with a granular medium and rotated around a horizontal axis. It has been extensively used to study the flow of dry materials (Jaeger et al., 1989; Rajchenbach, 1990; Liu et al., 1991; Evesque, 1991; Hill and Kakalios, 1994; Nakagawa et al., 1993; Ristow, 1996; Yamane et al., 1998; Dury et al., 1998; Gray and Tai, 1998; Bonamy et al., 2001; Orpe and Khakhar, 2001; Gray, 2001). The overall dynamics of the material surface in a rotating drum is a stick-slip process. At low rotation rates, the medium remains at rest relative to the drum while its surface angle is slowly increased by rotation, up to a critical angle (max ) where an avalanche occurs, thus decreasing the surface angle to the repose angle (r ). The flow becomes continuous at high rotation rates, but the transition between avalanching and continuous flow in dry media is hysteretic in rotation rate (Rajchenbach, 1990). To characterize the gross dynamics, we investigate the aforementioned hysteretic transition as a function of liquid content. We interprete some of our findings in terms of a simple stick-slip model. Then we quantitatively investigate the flow dynamics at different liquid contents by analysing the time evolution of the averaged surface profile obtained from hundreds of avalanche events and also by measuring surface velocities during continuous flow. In particular, we explore the nature of the viscoplastic flow, which

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CCD CAMERA

PRECISION STEPPER MOTOR ROTATING DRUM

COMPUTER

Figure 5.1: Sketch of the experimental setup. displays unique characteristics associated with coherent motion over the entire granular surface. Earlier studies of avalanches and surface flows in granular media have largely focused on dry grains (see Chapter 2). On the other hand, geological experiments on avalanches usually investigate very high liquid contents (Iverson et al., 2000). Our liquid contents are between the two extremes and we are thus able to investigate the gradual change in the behaviour. Experiments in our wetness regime were performed only recently (Bocquet et al., 1998; Fraysse et al., 1999; Nase et al., 2001), and these studies focused on the increased stability of the granular media at rest or on the size distribution of the avalanches (the difference between max and r ) (Quintanilla et al., 2001). In our measurements, we focused instead on characterizing the dynamics of wet grains, and in particular on how the cohesiveness associated with wetting leads to novel correlated behaviour - i.e. grains moving in clumps rather than individually.

5.1 The Rotating Drum Experiment We studied glass spheres thoroughly mixed with small quantities of hydrocarbon oil. The viscosity of the oil used was 0:27 poise, and its surface tension was 2 10 4 J/cm2 . The

liquid content varied between = 0:001 % and 5 % of the void volume. In this regime the flow of oil due to gravity can be neglected. The glass spheres were of the same type

as the ones described in Chapter 3. Measurements were performed on two sizes of beads, with diameters d1 = 0:9 mm 11 % and d2 = 0:5 mm 20 %. The results were usually 66

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qualitatively equivalent in the two cases, but we have found some differences too, as will be described later. The drum was made of thick plexiglas, but the vertical walls were lined with glass plates in order to minimize electrostatic effects. To prevent slips at the circumference of our drum we placed a hollow, thin aluminum cylinder into the drum, which was roughened at the inner side. The inner diameter of the drum was 16:8 cm, its width was 3:2 cm, and the granular filling was 30 %. By performing measurements in a thinner (2 cm) drum we verified that, while wall effects are not negligible, they do not modify the qualitative behaviour. The drum was rotated by a precise stepper motor with built-in gears (1 : 36), providing an extremely stable rotation rate that could be varied in the range of 0:003 30

rotation per minute (rpm). The stepsize of the drum was 0:025 Æ . The vibrations originating from the stepper motor were damped by many heavy lead bricks fixed to the apparatus. The stepper motor was controlled by a PC, which emitted impulses at regular intervals through the parallel port. These impulses were amplified by an operational amplifier and served as input for the driver circuit of the stepper motor; the motor performed a step for each impulse. A Turbo Pascal program was used to control the rotation rate. It automatically performed a series of measurements with different rotation rates according to a parameter file, this way the effect of rotation rate could be quickly and conveniently explored. The experiment was recorded by a CCD video camera interfaced to a computer which could analyse the spatiotemporal evolution of the surface profile (height variations in the axial direction were negligible). The steps of image processing are demonstrated in Figure 5.2 a-c. When the grains are wet, then the beads sticking to the glass walls make the surface hardly visible (Fig. 5.2 a). We overcame this problem by using background illumination (Fig. 5.2 b) and applying a brightness threshold (Fig. 5.2 c) on the image. Then we found the surface profile h(x) by an algorithm based on the detection of points

of maximum contrast (see Appendix B.1 for more details). The resulting surface data is presented in Figure 5.2 d-f in different ways. Fig. 5.2 d shows a test of the algorithm: the detected surface is superimposed on the photograph of the system as a thin black like. The limitations of our camera and computer gave us a resolution of Æt = 0:03 seconds and Æx = 0:5mm. Fig. 5.2 e presents the h(x) curve together with a straight line that was fitted to the curve in a rotation-invariant way: the squared sum of the geometrical distances of the surface point from the line was minimized. The inclination angle of this

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a

b

c

d

e

f

Figure 5.2: Steps of image processing. (a) Snapshot taken without background illumination. The wet grains sticking to the glass walls make the surface profile hardly visible. (b) The same picture but with a circular lamp turned on in the background. In diffuse background illumination the monolayer of beads on the wall is much brighter than the 3:2 cm thick sample. (c) A brightness threshold is applied to the image to make the beads on the walls disappear. (d) Test of the algorithm: the surface data (thin black line) superimposed over the original photograph. (e) The surface profile with a fitted straight line determining the overall surface angle ( ). (f) Pseudo-3dimensional representation of the surface profile, shading indicates local angle ((x)). This shading may be used to represent a given surface profile in a single line (see the horizontal stripe at the bottom). line gives the overall angle of the surface. This is the generalized version of the surface angle that has generally been used to describe the straight surface of dry samples. Finally, Fig. 5.2 f shows the surface profile with a brightness-coded representation of the local d h(x) ) in the third dimension. The stripe at the bottom is a angle ((x) = arctan dx concise way of displaying this surface profile using the same shading. Later we’ll use this type of representation to visualize the time evolution of the surface by placing the stripes corresponding to successive surface profiles under each other.

5.2 Measurements of the surface angle The phenomenology of our wet materials is much richer than the one observed with dry samples. The attractive forces between the particles lead to correlated motion of the 68

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surface angle (degrees)

a

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120 100 80 60 40

20

20 0

2

4

6 8 10 time (s)

12

14

0

50 100 150 rotation angle (degrees)

200

Figure 5.3: The temporal variation of the surface angle at various rotation rates (a) as a function of time, (b) as a function of the rotation angle of the drum. These data were taken using the large d = 0:9 mm beads, the liquid content was = 0:009 %. The rotation rates were 0:6 rpm (+), 0:75 rpm (), 1 rpm (), 1:2 rpm (), 2 rpm (), 3 rpm (Æ), 4 rpm (), and 6 rpm (4). Note that the curves are shifted by integer multiplicatives of 20 Æ for clarity. grains and qualitatively new avalanche dynamics. These new types of avalanches form rough, sometimes exotic surfaces. However, in order to capture the most fundamental features of our system, as a first step we neglect these surface features, and we use the overall surface angle (Fig. 5.2 e) to describe the observed surfaces with a single number. This approach allows us to compare our results directly to the behaviour of dry materials, where the surface is practically a straight line, being completely described by its inclination angle.

5.2.1 Flow Phases Figure 5.3 shows the time evolution of the surface angle for various rotation rates. Similarly to the dry materials we observe discrete avalanches at low rotation rates. (t) in this regime is a typical sawtooth-shaped function, which looks very similar to the fluctuations in the granular drag force (see Fig. 3.2 a) and to force fluctuations in other stick-slip processes. We can thus use the same algorithm to determine the maximum and minimum points now corresponding to max and R respectively. Note, that if we plot the surface angle as a function of the rotation angle of the drum, then the increasing segments of the

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50

2500s 1000s 500s 250s 100s 75s 50s 30s 20s

Surface angle (degrees)

45 40 35 30 25 20

0

0.2

0.4

0.6 Time (sec)

0.8

1

1.2

Figure 5.4: The variation of the overall surface angle during individual avalanche events for various rotation rates. The parameters of the sample are: d = 0:9 mm, = 0:187 %. curve are straight lines with a slope of unity. If we gradually increase the angular velocity of the drum, then at a given rotation rate ( = 4 rpm in Fig. 5.3) the flow becomes continuous, and after some transients the surface angle becomes constant.

5.2.2 Variation of the Surface Angle during an Avalanche The resolution of our measurement technique allows us to investigate in detail the variation of (t) during an avalanche. Figure 5.4 displays (t) during an avalanche for various rotation rates between = 0:02 rpm and = 3 rpm ( = 0:0625 %).

Our most important observations are the following: if is increased (i) max becomes smaller, (ii) r becomes larger, (iii) the slope of (t) decreases, the avalanches become

slower, (iv ) the total duration of the avalanche increases, and finally (v ) at = 1 a transition to continuous flow occurs. As we will discuss later (in section 5.4.2), the decrease of max (i) is related to the logarithmic strengthening of the material at rest. The other observations are easy to understand. If max and thus the potential energy of the initial state is smaller, then the kinetic energy of the avalanche will also be smaller, meaning a slower decrease of the angle (iii). When the angle decreases below a critical value (usually called the neutral angle

0 ), then the energy dissipation of the grains due to friction and inelastic collisions will be 70

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larger than the kinetic energy gain due to gravitational acceleration: the avalanche starts to slow down. A lower kinetic energy is dissipated more quickly; slower avalanches are stopped at larger angles (ii). The increase of avalanche duration for smaller avalanches (iv ) can be regarded as counter-intuitive, but considering that the decrease of the angle is slower, it is not very surprising. Also the origin of the transition to continuous flow (v ) is clear. As first approximation we can say that the transition occurs, when the time between successive avalanches becomes less or equal to the avalanche duration and so the avalanches merge. (Though, as we have seen, the avalanche duration depends on , so one has to be careful, when estimating the transition point.) These arguments may capture part of the truth, but they are quite wordy and imprecise. In the following section we show that all these observations can be easily interpreted in the framework of a simple stick-slip model.

5.3 Stick-slip Model for the Avalanches We emphasized many times that granular avalanches in a rotating drum are analogous to a stick-slip system, stick-slip models have been used to describe avalanches in a rotating drum by several authors (Benza et al., 1993; Caponeri et al., 1995; Linz et al., 1999). In this section we demonstrate this analogy. First we present a simple phenomenological argumentation that connects the dynamics of our system to the equations describing the stick-slip model, then we compare the predictions of the model to our experimental data.

5.3.1 Definition of the Model Like in the previous section, we average over all the microscopic processes taking place on the level of individual grains, and describe our surface by the overall surface angle

(t). The variation of (t) is determined by the interplay of the rotation of the drum and the flow of material, thus

d (t) = !J (t) + ; dt

(5.1)

where !J (t) is the change of the angle due to the flow (!J (t) < 0), which is proportional to the material flux, is the angular velocity of the drum. Now we have to specify the dynamics controlling ! J (t). As a starting point let us consider the steady

state, where (t) = S:S: = const, thus !J (t) = !0 = (this corresponds to the continuous flow regime). The steady state value of the surface angle S:S: depends on . There 71

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exists an inconsistency in the experimental observations (Rajchenbach, 1990; Lemieux and Durian, 2000) about the exact form of S:S: ( ), but if we assume that dissipation is larger at higher flow rates then it must be a monotonously increasing function. Our first important assumption is that this dependence is linear, i.e., we neglect the higher order terms in the power expansion: S:S: ( ) = 0 + a0 , where a0 and 0 are constant, positive

parameters. Thus the steady state flow rate !0 angle:

= is also a linear function of the surface

!0 () = a (

0 );

(5.2)

where a = 1=a0 . If the material flow rate is smaller than the steady state value, then the moving grains will accelerate and some of the grains at rest will be dislocated, so that

the flow rate will be increased. On the other hand, if j!J j > j!0 j, then the flow rate will decrease due to the increased energy dissipation. Again with linear approximation we obtain the closing equation of our model:

d ! = dt J

0

b(!J

!0 )

if !J = 0 and otherwise.

< max

(5.3)

Here b is a positive parameter, and we have added the important physical constraint,

that the material cannot flow uphill, ! J 0 always holds. If !J becomes zero, thus the flow stops, then !J remains zero, and (5.1) ensures that the is increased by solid body rotation until it reaches max . Then the flow starts again. We note that at low surface angles !0 may become positive. Though in this case it cannot be interpreted as the steady state flow, we get no unphysical behaviour, its only consequence is a quicker energy dissipation in (5.3). Combining the 3 equations of our model and eliminating ! 0 and !J we obtain:

=

ab

0 if _ = and < max :

_ 0 a b otherwise

(5.4)

This equation is fully similar to the stick-slip model with a friction force linearly increasing with velocity 2.11. We recall that this equation described a solid block pulled by a spring at velocity vs , in a coordination system also moving with v s . The rotation of the drum with an angular velocity corresponds to the flat surface under the block moving at a velocity vs in the moving coordinate system. In the stick phase the angle is increased by solid body rotation, slip events are the avalanches.

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During the avalanches (_ < ) the system behaves like to a damped harmonic oscillator. Then eqn. (5.4) can readily be solved in the traditional way by assuming that the solution has the form solution:

if

(t) = A1 e1 t + A2 e2 t . This way we obtain the following

b2 > ab 4 "

(t) =

r

A sinh

+ 0 + ; a if

b2 < ab 4

(t) = A cos

b2 4

!

r

ab t + B cosh

!#

b2 4

ab t

e

b

2

t

+ (5.5)

!

r

ab

b2 t + Æ e 4

b

2

t

+ 0 + : a

(5.6)

The overdamped solution (5.5) is not relevant to our system, so we’ll have to choose our parameters so that b2 =4 < ab. Then the slip event is described by eqn. (5.6), we only have to adjust A and Æ to the initial conditions:

(t0 ) = max _(t0 ) = :

(5.7)

This can be done easily and we obtain: 0

Æ =

arctan @

+ (max

(max ( ) A = max 0 a cos(Æ )

0

)

1

0 a A q

) ab b2 a 4

We have to keep in mind that eqn. (5.6) holds only while

(5.8)

(5.9)

_ < . When _ becomes

equal to , then eqn. (5.4) prescribes = 0, the avalanche stops. Solving the equation _ = is the only point that we cannot handle analytically in our model.

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2500s 1000s 500s 250s 100s 75s 50s 30s 20s


45 40 35 30 25 20

0

0.2

0.4

0.6 Time (sec)

0.8

1

1.2

Figure 5.5: The variation of the overall surface angle during individual avalanche events for various rotation rates. Symbols show the data presented in Fig. 5.4, continuous lines are fits of the stick-slip model.

5.3.2 Comparison to Experiments Now we compare the predictions of our model to the data presented in Fig. 5.4. Our first observation was that max strongly depends on the rotation rate . This is a consequence of an aging process that takes place while the material is at rest and this process is not influenced by the dynamics of the avalanche. Thus in our model we treat max ( ) as a given initial condition, for each rotation rate from the data in Fig. 5.4. We practically treat max as a free parameter for each series of data. (Note that the maximum points of the data series are not equal to max in our model, since after the avalanche is initiated, the angle still increases for a while due to the rotation.) However, all the other free parameters of the model: a, b and 0 are assumed to be independent of the rotation rate. We thus have 12 free parameters for the 9 curves. The result of the fit is presented in Figure 5.5. Given the crudeness of our model, the agreement is surprisingly good. Our model reproduces all the qualitative statements listed in sec. 5.2.2. We introduced that max ( ) is a decreasing function, and the model reproduced the increase of the minimum angle and the avalanche duration and the decrease of the slope of the curves. These four quantities are plotted explicitly in Figure 5.6. The model correctly reproduces the trends in all the quantities, and except for the 74

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Figure 5.6: Various parameters calculated from the avalanche data in Fig. 5.5 (symbols) and the corresponding fits (solid line). (a) Maximum angle max . Note that this quantity was treated as a free parameter, consequently the agreement is almost perfect. (b) Repose angle R . (c) Avalanche duration. (d) The maximum value of jd=dtj which is proportional to the maximum flow rate. avalanche duration (Fig. 5.6 c) its predictions are in quantitative agreement with the experimental data. This is strong evidence that the rotating drum setup can be described as a stick-slip system. A remaining question is the transition to continuous flow. We have already shown in sec. 2.1.3, that this model is able to reproduce the transition from the intermittent to continuous motion. Now in Figure 5.7 compare the transition in the model to the experimental observations. Figure 5.7 a shows the (t) for a = 0:0187 % sample at

= 4 rpm. At t = 10 s the rotation rate is suddenly changed to = 6 rpm, and a transition to continuous flow occurs (Fig. 5.7 b). The continuous lines show the prediction of the stick-slip model. Nothing but the rotation rate, is changed at t = 10 s, and the transition is correctly reproduced. Furthermore, the damped oscillations after the change in the rotation velocity are also in agreement with our model (though the period of the experimental oscillation is increasing slowly in time, while the oscillation period is con75

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12

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16

18

20

Time (sec)

Figure 5.7: Dynamics of the transition to the continuous flow. (a) At = 4 rpm the motion is intermittent. (b) At t = 10 s the angular velocity of the drum is suddenly changed to = 6 rpm, and after some transients the flow becomes continuous. Symbols: experimental data with the following parameters: d = 0:9 mm, = 0:0187 %. Solid line: fit of the stick-slip model. stant in the model). At first sight it might be counterintuitive to describe our experimental system as an oscillator, but Fig. 5.7 indicates that indeed, it may produce oscillations. Naturally, our model is a very basic one, and there are some trivial extensions which we are currently working on. The simplest improvement we can make is to write an empirical model for max ( ) that is able to reproduce the data in Fig. 5.6 a, this drastically reduces the number of free parameters without considerable loss of precision. The principal weakness of the current version of the model is that in eqn. 5.2 the dissipation was implicitly assumed to monotonously increase with the flow rate, and thus our model cannot reproduce the back-transition: the steady state solution is always stable. A straightforward solution to this problem is using the friction law of Jaeger et al. (1990) that has a minimum as a function of the velocity (see eqn. 2.15), and writing a modified stick-slip model as done by Benza et al. (1993). This approach has the additional benefit that the parameters of the models will be related to processes on the grain level. We can then attempt to set up relations connecting the parameters to the liquid content of the sample and reproduce the phase diagrams presented in section 5.4.

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ω = 3 rpm

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f

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Angle (degrees)

Rotation Rate (rpm)

10

Rotation rate (rpm)

50 40 30 20 10 1E-3

0.01

0.1

1

Liquid content (% void space)

Figure 5.8: (a) Phase diagram of the dynamical behaviour as a function of the liquid content ( ) and the rotation rate (! ). (b, c, d) max and r as a function of rotation rate for 3 different liquid contents as indicated by the white arrows in the phase diagram. The rotation rate was first increased and then decreased adiabatically. (e, f) max and r as a function of liquid content for different rotation rates.

5.4 The Phase Diagram We then investigated the avalanching – continuous flow transition for various liquid contents. We slowly increased and then decreased the rotation rate and measured the time-dependence of the average surface angle, calculating max and r . The order parameter determining the type of flow is = hmax i hR i (averaging is for independent avalanches). In the continuous flow regime is close to zero, in case of discrete avalanches it has a finite value. The results of typical runs are presented in Fig. 5.8 b, c and d, where max and r are plotted as a function of the rotation rate. A summary of all our measurements of this type is shown in Fig. 5.8 a indicating the nature of flow as a function of ! and like a phase diagram. Red diamonds indicate the sets of parameters where is larger than a threshold (1Æ ), i.e. the flow is avalanching. Blue symbols mark the places where the order parameter is always smaller than the threshold, the flow is con77

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tinuous. The white diamonds correspond to the hysteretic region: if we slowly increase the angular velocity of the drum, then the flow remains avalanching until we reach the upper edge of the white region (the upper edge corresponds to 1 ( )). On the other hand, if we slow down the quickly rotating drum exhibiting continuous flow the transition to avalanches occurs only at the lower edge of the region (corresponding to 2 ( )). Fig. 5.8 a demonstrates that the three regimes of behaviour observed previously are reflected in the dynamics of the present experiment. In the granular regime at low oil contents, ( 0:015 %), the behaviour is qualitatively similar to the dry case, and we observe a clear hysteretic transition between the continuous flow and avalanching (Fig.

5.8 b). The correlated regime at intermediate oil contents (0:015 % < < 0:08 %) is marked by the lack of hysteresis (Fig. 5.8 c), and displays avalanching behaviour at a relatively high rotation rate compared to the other regimes. The correlated motion of grains in a clump is an effective form of transport, avalanches have a much shorter duration, t1 , in this regime (Tegzes et al., 1999). This explains why the transition point

1

(max R )=t1 is at relatively high rotation rates.

Furthermore, even when the time interval between successive avalanches decreases below the avalanche duration, we see no qualitative change in the nature of the flow. The continuous flow in this regime consists of a stream of separated clumps rather than the constant flux seen in the other two regimes. This accounts for the lack of the hysteretic transition. Further increasing the rotation rate we enter the S-shaped regime and the clumps break up due to inertial effects. The sudden change in the transition point 2 near = 0:1 % reflects the onset of the viscoplastic regime. It reveals that the smooth viscoplastic continuous flow can be maintained until much lower rotation rates than the collisional flow of the granular regime. Here we observe hysteresis again (Fig. 5.8 d). If we observe the behaviour as a function of liquid content (Fig. 5.8 e-f) then we can see different scenarios depending on the value of the rotation rate. At low rotation rates we always observe avalanches (Fig. 5.8 f). Here max ( ) reflects the changes in the stability of the material due to the formation of liquid bridges. This curve can be compared to Figure 4.2 (note that the horizontal axis is logarithmic in Fig. 5.8 f). At very high rotation rates the flow is always continuous. However, as a direct consequence of the properties of the phase diagram, at intermediate rotation rates we can observe 2 transitions: at 0:01 % from the (collisional) continuous flow to avalanches and then at

0:08 % back to continuous flow (which has viscoplastic properties) (Fig. 5.8 e).

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Liquid content (% void) Figure 5.9: Shape of the surface at various points of the parameter space. We note that Fig. 5.8 f displays no change in behaviour at

0:08 %, but there is

a sudden decrease of max near 0:5 %. Visual observation reveals that this decrease corresponds to the onset of smooth viscoplastic avalanches. The boundary between the 3 flow mechanisms thus cannot be approximated by vertical lines in the phase diagram. Clearly, Fig. 5.8 a is insufficient to characterize the dynamical behaviour: though the changes in the transition points reflect the different flow mechanisms, we would need other order parameters which distinguish between granular, correlated and viscoplastic flows.

5.4.1 Surface Shape One candidate for such an order parameter can be the shape of the surface (Fig. 5.9). In the granular regime the surface is almost straight, correlated avalanches form a typically

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Figure 5.10: Shape of the surface at various points of the parameter space. concave surface, and the viscoplastic flow corresponds to a convex surface. At the highest rotation rates the surface is S-shaped due to inertial effects across the entire wetness range. The convexity of the surface can be characterized quantitatively several ways. A simple and robust measure is the signed area between the surface line and the straight line connecting the two ends of the surface line (). For a straight surface means a predominantly concave, > 0 means a mostly convex surface.

= 0, < 0

In Figure 5.10 we the value of is indicated in the colour of symbols as a function of and . Black symbols correspond to concave surfaces (negative ), and approximately

indicate the correlated regime. White symbols are used for relatively big positive values, corresponding to the viscoplastic regime. Grey symbols indicate that 0, either because the surface is straight (upper left region of Fig. 5.10 - this is approximately the

granular regime), or because convex and concave parts are balanced (at the boundary of correlated and viscoplastic regimes).

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Figure 5.11: The dependence of the maximum angle max on the rotation rate . The system exhibits aging: longer waiting times between avalanches leads to increased stability. The continuous lines are logarithmic fits to the data.

5.4.2 Aging The fact, that changing the rotation rate can switch between different flow mechanisms indicate that the formation of correlated clumps is a dynamical process with characteristic times comparable to the other timescales of the experiment. A related interesting feature is observable in the curves of Fig. 5.8 b-d. Although (at slow rotation) the rotation rate only influences the waiting time between successive avalanches, max decreases with increasing rotation rate. The curves indicate a logarithmic increase in the stability of the system as a function of waiting time. This effect is presented in more detail in Figure 5.11, which plots the max as a function of ). Similar aging effects have been observed in several different granular experiments (Bocquet et al., 1998; Losert et al., 2000). The underlying mechanism is debated, several factors may play an important role (see sec. 2.2.3). Since in our experiment the effect is more pronounced at higher liquid contents, and the vapour pressure of oil is low, we attribute it to the motion of oil flowing towards the contact points rather than condensation effects (Bocquet et al., 1998). This 81

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Figure 5.12: The oil at the contact point of a wet bead (similar to the ones used in the experiment, but d = 1:2 mm) and a glass plate at different time instants observed via phase-contrast microscopy. The delay time from the instant when the bead was placed on the plate was (a) t 5 s, (b) t 15 s, (c) t 25 s. The flow of oil to the contact point occurs on a timescale of several seconds. assumption is also supported by our investigations of a wet contact by a phase contrast microscope. We placed a wet bead to the object plate and observed the development of the oil interface at the contact point (Fig. 5.12). Our observations revealed that a liquid bridge was formed instantaneously, but then its shape was changing slowly, and we still observed noticeable changes after several seconds of waiting time.

5.4.3 Effect of Parameters on the Phase Diagram If we want to draw general conclusions from our phase diagram, we have to investigate how universal our it is, i.e. to what extent it changes if we change the parameters of the system, like the geometry, the granular medium or the interstitial liquid. We therefore performed some comparative measurements with beads of different size: d2 = 0:5 mm 20 %, d3 = 0:35 mm 20 % (Fig. 5.13). These diagrams contain fewer points, but the main trends are visible. A universal feature of the phase diagrams is that approaching the correlated regime from the granular one the transition point moves towards higher rotation rates — the clump formation leads to a more effective transport and quicker avalanches. Also, for smaller beads the transition to correlated behaviour occurs at higher liquid contents, most probably due to the larger internal surface of these samples. As the bead size is decreased, the viscoplastic regime, and in particular the viscoplastic continuous flow seems to diminish. The

0:5 mm beads exhibited continuous viscoplastic flow only in a narrow region, 82

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Figure 5.13: Phase diagrams for smaller beads. The bead diameters are: (a) d2 = 0:5 mm, (b) d2 = 0:35 mm. Notations as in Fig. 5.8. The 0:5 mm beads exhibit viscoplastic continuous flow in the white region near 0:4 % and 1 rpm if the velocity is gradually decreased. for the the smallest beads such flow was not observed at all. In samples consisting of smaller beads the number of contacts at the boundary of a given volume of material is much larger, therefore the cohesive forces dominate inertial effects. We also observe a tendency that for smaller beads the hysteresis of the continuous-to-intermittent transition seems to decrease. This is hard to interpret and would require further investigations. We have performed some preliminary measurements on the role of the viscosity of the interstitial liquid. The results (not shown here) indicated that the fluid viscosity is indeed an important parameter. E.g. experiments with a highly viscous silicon oil yielded viscoplastic flow in a wide range of liquid contents and rotation rates.

5.5 Avalanche Dynamics By using the rotating drum apparatus, we can obtain information not just about the medium before and after the avalanche events, but we can also study the details of the grain motion during avalanche events. In order to analyse the dynamics of avalanches we have obtained two dimensional space-time matrices, h(x; t), characterizing the sample surfaces throughout the avalanche process which can then be analysed to produce

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a variety of information about the individual avalanches. By taking derivatives of the h(x; t) data, we obtain the local angle ((x; t) = arctan[@x h(x; t)]) and the local verti-

cal velocity (u(x; t) = @t h(x; t)) of the surface profile. Furthermore, by then integrating the vertical velocity (using the continuity equation and assuming constant density (Douady et al., 1999)), we also obtain the local flux in the avalanche, i.e. (x; t) =

w

Rx 0 0 D=2 @t h(x ; t)dx (where

is the grain density, w is the width of the drum, and D

is the drum diameter), which represents the material flowing through a given vertical plane at position x. Since the observed features are more robust for the smaller beads (d = 0:5 mm), in this section we present data obtained using these beads. In subsection In Fig. 5.14 (a) we present snapshots of the progression of single avalanches for several typical liquid contents. Fig. 5.14 (b) displays (x; t) as a function of space (horizontally) and time (downwards) for the same individual avalanches. In Fig. 5.14 (c-e), we present the average behaviour of 300 500 avalanches at the same liquid contents, and show similar graphs of the time evolution of h(x; t)i, hu(x; t)i, and h(x; t)i, where hi denotes averaging over avalanches. By obtaining these quantitative measures of the averaged properties, we can separate the robust characteristics of the avalanche dynamics from the large fluctuations which are inherent in avalanche processes. In Figure 5.15 we present the same data as series of curves. The columns represent different liquid contents. In the first row of graphs we plot the averaged surface profiles(h(x)i) at different time instants during the avalanche. The second raw consists

of graphs showing the variation of the local angle (h(x)i), the third one contains graphs of the local flux ((h(x)i)). The purple triangles show the position of maximum flux. In the following we analyse the dynamics in the 3 regimes based on Figs. 5.14 and 5.15.

5.5.1 Avalanche types Granular avalanches. In the avalanches among dry grains (Rajchenbach, 1990; Liu et al., 1991; Jaeger et al., 1989; Daerr and Douady, 1999b; Bretz et al., 1992; Frette et al., 1996), the surface remains almost straight throughout the avalanche, and the avalanches have a much longer duration and much smaller flux than in the wet media - as is expected due to the lack of cohesion. Our resolution is not enough to distinguish any propagating front in this case. Our system is very similar to the one investigated by Rajchenbach (2000), all the surface grains are close to the limit of their stability. Thus we assume that the failure mechanism is similar: the propagation of the front destabilizing the grains is quicker than 84

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Figure 5.14: Dynamics of avalanches of different types with grain size d = 0:5mm. (a) Snapshots (at 0.1 – 0.2s time intervals) of four single avalanches corresponding to different liquid contents. The third dimension is used for a brightness-coded representation of the local slope (). The green lines indicate approximate slip planes in the correlated regime, and the red line shows the travelling quasi-periodic surface features in the viscoplastic regime. (b) The local slope with the same brightness coding, as a function of space and time. A horizontal line corresponds to a surface profile at a given instant, ! 85

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the material flow (conf. sec. 2.3.2). This type of front is marked only by a very slight dilation of the material, which we cannot detect. At low but nonzero liquid contents (e.g. = 0:04 %) the avalanches become much larger due to the onset of intergrain cohesion — this allows us to observe the dynamics in more detail. In this granular regime the avalanche is always initiated at the top of the surface, and the upper part of the surface is quickly destabilized. As the particles start moving downwards, a front of rolling grains travels downhill. The grains at lower regions remain at rest until the rolling grains reach them. This type of front corresponds to the “start down” front of Douady et al. (2001, see also sec. 2.3.2). The difference in behaviour compared to the perfectly dry material can probably be explained by the slight concave curvature of the surface: the grains at lower regions are in a relatively stable state when the avalanche is initiated — similarly to the situation in the experiments of Daerr and Douady (1999b). When the downward-propagating front reaches the wall of the drum at the bottom of the slope, a kind of shockwave is formed, and the rolling of grains is stopped by a region of smaller local angle travelling uphill. The second front is classified as a “stop up” front by Douady et al. (2001), and corresponds to the “kink” seen in other experiments on dry beads (generally associated with individually rolling grains reaching a solid barrier at the bottom of the slope (Makse et al., 1997; Gray and Tai, 1998; Cizeau et al., 1999)). The different front types observed in our setup, are visible in Figs. 5.14 and 5.15 (see captions for a detailed explanation). For clarity, we have magnified some parts of these figures in Figure 5.16. Correlated Avalanches. At intermediate liquid contents, in the correlated regime (0:1 % < < 2 %), the principal failure mechanism is a fracture along a curved slip plane (approximated by the green lines in Fig. 5.14 (a)), analogous to the dynamics of a

Figure 5.14: (cont.) and time increases downwards (thus avalanches propagate down and to the left). The slanted bright and dark regions correspond to the avalanche front and the kink respectively (see text). The stripes at higher liquid contents indicate lasting surface features. (c) (d) (e) The characteristic features of the avalanches averaged over 300 500 avalanche events. The displayed quantities are (c) the local surface angle, (d) the rate of change of local height, and (e) the local grain flux (the white lines indicate the point of maximum current), respectively. Note that the surface patterns at the highest oil contents are robust against averaging.

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[5

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Figure 5.15: Averaged dynamical parameters: hh(x)i, h(x)i, and h(x)i as a function of space at various time instants during the avalanche. The time interval between successive curves is 0:066 sec, some curves are marked with colour as specified in the legend. In the lowest row triangles mark the point of maximum current. The most important features for different oil contents are the following. Dry samples. The avalanche is very small, our resolution is insufficient to resolve the detailed dynamics. = 0:04 % Granular regime. The surface is close to straight, but the local angle is somewhat higher near the top. The dynamics is dominated by fronts of rolling grains. The region of steeper slope corresponding to the downward propagating front as well as the negative slope of the kink is visible in (g). The point of maximum current also moves downhill then uphill due to the fronts. = 0:12 % Correlated regime: failure occurs along a slip plane and a block slides down. The block is best seen in the red curve, the edges are smoothed by the averaging over several hundred avalanches. Some rolling grains are still present, as marked by the presence of the kink. = 0:37 % Correlated regime, usually with multiple avalanches, but these are averaged out. The fronts have disappeared, the angle near the bottom of the slope decreases monotonously. There are lasting contacts in the avalanche, the grains cannot roll, the material stops coherently as a block when it reaches the bottom. = 5:00 % Viscoplastic regime. The surface moves coherently. As presented in (o), the current extends over the whole surface during the whole duration of the avalanche. Curves of the local angle in (j) are even more revealing. The robust surface patterns are clearly visible. During the avalanche the local maxima move downwards (and spread out slightly), and a new maximum is formed near the top.

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Figure 5.16: The various fronts observed in our setup in the granular regime ( = 0:04 %) and in the low liquid content region of the correlated regime ( = 0:12 %). (a-b) The rate of change of local height during the avalanche, (c-d) the spatial dependence of the local angle at various time instants. We observe 4 different fronts. 1: destabilizing front. Propagates faster than the material flow via interactions between neighbouring grains — destabilizes particles. 2: Avalanche front. Consists of rolling grains, which proceed on a medium at rest and destabilize particles by collisions. 3: Kink. The rolling grains accumulate at the lower wall and form a region of low slope there. This slows down the incoming grains, thus this front travels uphill. 4: Rear front. The rolling grains leave the static ones behind. class of geological events known as “slides” (Dikau et al., 1996). This is the failure mechanism described by the Mohr-Coulomb model (see sec. 2.2.2). A = 0:12 % there is a single slip plane, but at larger liquid contents ( = 0:37 %) the avalanches occur through a succession of local slip events. The medium becomes more cohesive with increasing liquid content, the rolling of grains becomes impossible and thus the kink disappears for 0:3 %. The material moving as a connected block stops coherently when it hits the bottom. Viscoplastic Avalanches. At the highest liquid contents, the onset of the viscoplastic regime ( = 2 %) is accompanied by dramatic changes in the behaviour. The flow becomes correlated across the entire granular surface as demonstrated by the parallel lines 88

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during the avalanche in Fig 5.14 (b) and (c). Since the whole surface moves coherently (rather than breaking apart and evolving separately in different parts of the surface), fluctuations are strongly suppressed (Tegzes et al., 1999). This behaviour is qualitatively similar to a different class of geological event, called a ”debris flow” or ”mudflow” (Dikau et al., 1996; Iverson, 1997; Coussot, 1997). The coherent nature of the motion in this regime leads to novel aspects of the dynamics which are not observed in any other granular system.

5.5.2 Pattern formation One novel property of the viscoplastic avalanches is the robust topology of the top surface which spontaneously forms a nearly periodic pattern (seen in Fig. 5.14 a and b). This surface structure is maintained essentially intact during the avalanche (note that the lines are continuous throughout the avalanche in Fig 5.14 b), indicating that there are lasting contacts in the flowing layer. Moreover, the pattern is not random, but rather has features which are reproduced at the end of each avalanche. This is demonstrated most clearly in Fig. 5.15 j and in Fig 5.14 (c), where the average of 347 avalanches of the = 5 % sample has the same features as the typical individual avalanche shown in Fig. 5.14 a and b. The robust nature of the surface structure of the wettest grains is in sharp contrast to the other regimes where averaging completely smoothes out the smaller surface features. We can understand this behaviour as resulting from coherence of the entire flow, which strongly reduces fluctuations in this regime. With minimal fluctuations, the final surface structure after each avalanche is essentially the same, thus setting the same initial condition for the next avalanche. With the same initial conditions for each avalanche, naturally the surface features are reproduced each time. Our experiments with the larger (d = 0:9 mm) beads also revealed some pattern formation (see Fig. 5.17 b), but with a smaller characteristic size corresponding to 8-10 grain diameters. The difference is probably due to the smaller ratio of the cohesive forces to the gravitational forces on the grains. The determination of the dependence of the pattern size on the size of the grains and the container and possibly other factors needs further investigations.

5.5.3 Larger Beads Apart from the robust large-scale patterns, the qualitative features of the avalanches in samples of larger (d

= 0:9mm) beads (see Fig. 5.17) are the same as the ones observed 89

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Figure 5.17: Avalanche dynamics in samples consisting of large beads d = 0:9 mm

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Avalanche energy (J) Figure 5.18: Distribution of the avalanche energy (difference between potential energy of the initial and final state) for various liquid contents. The inset shows the width of the peaks as a function of liquid content (full width at half maximum). for small beads, though the effects are less pronounced. In general we can say that while the details of the graphs presented in this section are determined by the geometry of the system and the properties of our samples, the 3 fundamental mechanism of avalanches, the granular, the correlated and the visco-plastic regime are generally applicable to wet granular systems.

5.5.4 Statistics of Avalanche Sizes We can better understand the evolution of the different avalanche regimes by considering the statistical distribution of avalanche sizes (the amount of dissipated energy) shown in Figure 5.18. Consistently with experiments in dry grains we found no self-organized critical behaviour, in most cases we observe Gaussian-like distributions around a finite value. However, the width of the distribution changes in an interesting way as function of liquid content. The width of the distribution is very narrow for dry grains, but also 91

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the avalanches are very small here, so the relative width of the distribution is fairly large. Around 0:1 % the relative peak with becomes very small, this is the region, where the dynamics is dominated by a single slip. At larger liquid contents the distribution width expands dramatically indicating the appearance of multiple avalanches. Here due to the clumping the system behaves as if it had only very few, large particles, and this can lead to a wide avalanche size distribution (Liu, Jaeger, and Nagel, 1991). For the very wet coherent flow the distribution becomes narrow again. In this wettest regime, the entire grain mass moves coherently (rather than breaking apart and evolving separately in different parts of the surface), and thus fluctuations in the dynamics are suppressed.

5.6 Continuous Flows In the previous section we demonstrated that avalanche dynamics is dramatically different in the three regimes of behaviour. The coherent motion of the grains in the viscoplastic regime contrasts sharply with the dynamics of separated clumps and individually moving grains at lower liquid contents. This difference has profound effects at higher rotation rates too, where the flow is continuous. In this section we compare the properties of the continuous flow in the three wetness regimes. Here again we use the large beads, since the viscoplastic continuous flow appears in a much wider region in their parameter space.

5.6.1 Velocity at the Surface Figure 5.19 shows the evolution of the surface profile during different types of continuous flow ( = 10 rpm, the drum is filled to 30 %). In the granular regime (e.g. = 0 and

0:002 %) the surface is practically a straight line with only noise-like variations. Our resolution is insufficient to resolve the individually rolling grains. In contrast, in the

correlated regime the surface is clearly concave, and the flow consists of a succession of clumps moving downhill. From the slope of the stripes in 5.19 b we can estimate the velocity of the moving clumps: v 50 60 cm/s. In the viscoplastic regime the flow is smooth again, and the travelling surface features testify that there are lasting contacts in the flowing layer. The stripes indicate that the flow is much slower: v 10 15 cm/s. We note that the rotation rate is the same for all the presented liquid contents, meaning that the material flux is equal too. The big difference in surface velocity probably indicates that in case of viscoplastic flow the moving layer is much deeper.

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Figure 5.19: Comparison of the types of continuous flow at = 10 rpm (large beads: d = 0:9mm). (a) snapshots at time intervals of 0:067 s. The red marks indicate travelling surface features. (b) time evolution of the surface angle (x; t). The travelling surface features appear as stripes. In order to compare the viscoplastic flow to the ordinary granular flow, we added some tracer particles to measure their surface velocity v more accurately as a function of rotation rate , (Fig. 5.20). The viscoplastic flow is slower by a factor of two than the granular one, and more importantly, the two curves correspond to rather different functional forms. The curvature of the granular curve suggest that at higher rotation rate the flow depth is increased as has been previously suggested (Bonamy et al., 2001). Naturally in our finite setup the flow depth cannot increase infinitely which may explain why our measured v ( ) deviates from 1=2 which would be expected from a linear velocity profile (Rajchenbach, 1998; Bonamy et al., 2001; Yamane et al., 1998). The linear v ( ) function of the viscoplastic flow suggests a constant flow depth which is independent of

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Rotation rate (rpm) Figure 5.20: The velocity v at the surface as a function of rotation rate for the ordinary granular and the viscoplastic continuous flow (large beads, d = 0:9mm). The continuous lines represent power-low fits to the data. The linear curve for the viscoplastic flow indicates that flow depth is independent of the rotation rate. the flow rate. This observation is consistent with the coherent nature of this type of flow: the flow depth is not determined by local mechanisms (Rajchenbach, 2000), but is fixed by the geometry of the whole system.

5.6.2 Flow Depth In order to characterize the viscoplastic flow in greater detail and check whether the flow depth is really constant we have performed explicit measurements to determine the extensions and shape of the flowing layer. A standard method for such measurements is to take two pictures shortly after each other, and analyse the differences between the corresponding pixels in the two images. In the static layer the differences are close to zero, while in the flowing layer large differences are observable. This works perfectly for situations, where the static layer is really at rest with respect to the camera. However, in our case the static layer undergoes solid body rotation, therefore also there the pixel values are changing. A simple rotation of one of the images cannot be used to solve the problem, e.g. because then the grid of pixels is rotated, thus we have to resample one of 94

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the images to make the comparison. This way the noise will be too large to get sensible results. To overcome the problem we used the following procedure. The drum was rotated at a constant velocity until a steady state was reached. Then we suddenly stopped the drum, and took two snapshots immediately after it (within 0:2 s). Since the steady state

angle during the flow is somewhat larger than R (and also due to inertial effects) the flow continued for a while (about 1 2 s). Now the static part was really at rest, so we could determine the flowing regions using the above described method. Naturally, the background illumination – used to produce sharp contrast at the surface profile – was replaced by a nearly homogenous lighting from the front. The results are presented in Figure 5.21. The first column shows the image differences for different rotational velocities: a: 2:4 rpm, b: 0:6 rpm, c:0:2 rpm, d: 0:06 rpm. Each image represents the average of 3 independent measurements. The bright region corresponds to the flowing layer. In the second column we superimposed the brightest pixels as black points over the original photographs. Taking into account that the flow rate in Fig. 5.21 a is 36 times larger than in Fig. 5.21 d, the difference in flow depth is rather small. In Fig. 5.22 plot the values of the measured depths as a function of rotational velocity. The change in flow depth is about 15 %, which lies within the uncertainty

of the measurement. The dashed line corresponds to h / 1=2 , which is consistent with a linear velocity profile with constant gradient, observed for dry granular flows.

5.6.3 Velocity Profile The most interesting but experimentally very difficult question about this novel type of flow is the shape of the velocity profile. Its significance lies in the fact that it provides direct information about the dominating dissipation mechanisms in the flowing region. The most difficult experimental problem is that the sample is opaque, and we can only observe the flow at the walls. This problem is particularly serious in our measurements, since the interaction with the walls is much stronger than in the dry case, and a monolayer of beads is actually sticking to the walls. Lacking a suitable MRI apparatus — which has been successfully applied to explore granular flow in 3 dimensions (Nakagawa et al., 1993)— we attempted to obtain valuable information from the boundaries. As a first step we analysed the flow on the free surface of the material. Fortunately the lasting surface structures in the relatively slow viscoplastic flow provide ideal conditions for determining velocities. We used an algorithm, which implemented a simplified 95

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b

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c

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d

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Figure 5.21: The moving layer in viscoplastic flow. (a) The pixel by pixel difference of two successive snapshots taken with a time difference dt = 0:03 s (see text for the details of the method). Bright pixels correspond to large differences, i.e., the flowing layer. The image shows an average of 3 independent measurements. The rotation rate was 2:4 rpm. (b-d) Similar plots for different rotation rates: 0:6 rpm, 0:2 rpm, and 0:066 rpm. (e-h) The brightest pixels from (a-d) respectively superimposed as black points over the original snapshots. Note, that the flow rate is 36 times larger in (a) than in (d), and there is only a relatively small change in flow depth.

96

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0.5

Relative Flow Depth (h/R)

0.4

0.3

0.2

0.1

0.0 1

2

Rotation Rate (rpm)

Figure 5.22: The depth of the moving layer as a function of the angular velocity of the drum in case of viscoplastic flow. The different symbols correspond to different methods of determining the depth from the images of Fig. 5.21. The dashed line corresponds to constant flow depth, the continuous line represents t / 1=2 which is consistent with the linear velocity profile observed for dry granular flows. version of particle imaging velocimetry, i.e. it determined local velocity from the displacement of local patterns between two consecutive snapshots (see Appendix B.2.1 for details). The obtained velocity filed is shown in Figure 5.23. The results indicate, that wall effects only play a role in the immediate vicinity of the walls (in a distance of 1 2 grain diameters), most of the material moves with the same velocity. This observation, combined with the fact that a few layers of glass beads are actually transparent allowed us to determine the velocity profile of the flow. We added some large (3 mm), pink, plastic tracer particles to the sample. The lasting contacts in the layer ensured that they moved at the same velocity as the surrounding glass beads, and their surface properties prevented them from sticking to the walls. These tracers were recognizable even if they were a few millimeters deep inside the material. We took colour pictures and calculated the ratio of the red colour band to the green and blue colours for each pixel. This quantity showed sharp peaks at the position of the tracer, and from tracer positions in successive images we could determine the velocity (the details of the algorithm are described in Appendix B.2.2). However, the large size of the tracers caused a serious problem: these particles tended to avoid regions of high 97

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Figure 5.23: Velocity field of the viscoplastic flow as obtained by particle imaging velocimetry. Note that the walls have little effect on the flow. shear, which prevented us from obtaining information about the most interesting parts of the sample. The obtained velocity field is presented in Figure 5.24 a. The bulk of the material undergoes solid body rotation, and a relatively thick layer is moving on the surface. The shape of the flowing layer is in agreement with the one observed via pixel by pixel image differences (Fig. 5.21). Similarly to the observations of the flow on the surface (Fig. 5.23) we find that the flow velocity in the flowing layer is remarkably homogeneous: the shear rate is close to zero! In order to further investigate this interesting phenomenon we have plotted the measured velocities vs. depth in Figure 5.24 b. In regions of no shear the measured velocity increases slowly with depth due to the rotation. This graph confirms, that in the top region the shear rate is zero. We can interpret this result in the framework of the Bingham model describing viscoplastic materials (see e.g. Johnson, 1970), which is a useful engineering approximation to describe debris flows (Huang and Garcia, 1997). In simple shear the stress and strain relation for a Bingham fluid is:

@v = @y

0 if @v yield sgn( @y ) if 98

j j < yield ; j j yield

(5.10)

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'HSWKFP

9HORFLW\FPV

Figure 5.24: (a) Velocity field of the viscoplastic flow as obtained by tracer particles. The central region is empty due to lack of data in this region. The actual resolution is much better than the separation of the arrows — only part of the data is presented for clarity. (b) Velocity profile along the thick solid line in (a). The diamonds are experimental data, the big open circle shows the flow depth obtained from Fig. 5.22. The solid line is a fit of the Bingham model to the data, dashed lines correspond to no shear strain. The arrows are only for illustration. where is the viscosity, v is the velocity of the material, is the shear stress, yield is the yield stress, sgn(x) = x=jxj is used for obtaining the correct sign. The meaning of this equation is that the material is able to sustain a finite shear force without deformation, @v ) and then above this threshold the shear stress has two components: a viscous one ( @y a plastic one yield = const. In Bingham materials the existence of a yield stress leads to an interface at which the shear stress equals yield . This interface divides the fluid into two parts: shear flow region

with varying velocity and > yield , and a plug flow region with uniform velocity and < yield . If a Bingham liquid is flowing on an inclined plane, and increases linearly with depth, then the velocity profile will be:

v0 if y < hplug (5.11) 2 A (y hplug ) if y > hplug ; where hplug is the width of the plug layer, A = v0 =(htotal hplug )2 , htotal is the total layer v= v 0

width. 99

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The viscoplastic flow in the rotating drum is more complicated, since there exists an additional interface too, between the flowing surface layer and the static bottom. The calculation of this second interface requires full consideration of the geometry of the system. However, if we fix htotal = 3:9cm (measured earlier, see Fig. 5.22), then our limited amount of data is in good agreement with (5.11), we show a fit in Fig. 5.24 b. Checking this theoretical curve for mass conservation reveals that it provides about 30 % larger total current than expected. This indicates that the major source of error is not wall friction, which would lead to an error with an opposite sign (Nasuno et al., 1998).

5.7 Discussion While the flows we observe appear to have analogies with geological events, it is important to note that our samples are quite different from real geological materials usually

consisting of polydisperse irregular particles with very high ( 100 %) water contents and that the scaling of our system to geological lengthscales is non-trivial. Interestingly we still recover some of the basic dynamical processes in our model system by tuning a single parameter, the liquid content. This constitutes an important step towards the description of qualitatively different flow behaviours in the framework of a single model (Nase et al., 2001; Gray, 2001). The changes in the dynamical behaviour are associated with the increasingly coherent nature of the flow, i.e. the formation of coherently moving clusters due to the increased cohesion and viscous effects. Within a cluster, local velocity fluctuations should be suppressed, and thus the local granular temperature

( T =< v 2 > < v >2 ) should approach zero, but the clusters themselves both form and break apart during an avalanche process in a finite container. An important theoretical question is how a length scale characterizing the size of this locally frozen volume of grains may emerge from a granular flow model, and how such a length scale would vary with the type of media, the nature of intergranular adhesion, and the type of granular flow.

100

Acknowledgement First of all I would like to give thanks to my advisor Tamás Vicsek for his constant support and encouragement, his profound comments and helpful advice. I learned from him to think and address problems as a physicist. He has created excellent conditions for scientific work always keeping my benefit the top priority, which enabled me to work according to the best of my ability. Right after him I have to express my gratitude to Prof. Peter Schiffer, my advisor during my staying in the USA, for his kind hospitality, and all his scientific and personal help. He taught me how to distinguish essential things and do experimental physics efficiently. There are many other people who helped me in my work. I am grateful to the students in Tamás Vicsek’s group: Zénó Farkas, Balázs Heged˝us, Illés Farkas, and János Kovács, and the students of Peter Schiffer: István Albert, Yee-kin Tsui, Steve Potashnik and Joe Snyder, who gave help every time I needed and whose kindness and friendship created a very pleasant atmosphere. I have to give special thanks to Zénó Farkas, who spent a lot of time helping me out in my struggles with the Linux operation system. I am indebted to András Czirok and Zoltán Csahók for the image processing routines they provided me at the beginning of my work and to Albert László Barabási for many fruitful discussions. I thank Sándor Hopp, Miklós Csiszér and Endre Berecz for their help in designing and preparing the experimental setups. I could not have completed this work without the kindhearted and generous help that my parents, the parents of my wife, and my brothers and sisters granted me in many fields of life. And finally, I am extremely grateful to my wife, Emese, who sacrificed most for the sake of this thesis, and always stood by me with her patience and love.

101

Appendix A Laboratory Exercise for Physicist Students This appendix briefly describes the laboratory exercise that I assembled for the “Modern Physics” laboratory course for physicist students (for a detailed description see (Tegzes, 2000)). The aim of this exercise was to give an introduction to the future physicists about the interesting phenomena and challenging theoretical questions presented by granular materials, and make them conscious of this novel field of modern physics. Due to our limited possibilities we decided to investigate the static properties, and chose experiments that were robust and required no sophisticated techniques. Nevertheless, the selected measurements are suitable to demonstrate the unusual properties of granular media at rest. We laid emphasis on two related fields: the internal static friction that is responsible for the non-ergodic, metastable nature of granular materials caused by and the lack of thermal averaging, and the she self-organizing network of grain-grain contacts that leads to the formation of force chains. The exercise therefore consists of two measurements, the first one investigates the pressure on the bottom of a granular column, the second one determines the distribution of forces on individual grains. A sketch of the experimental setup for the pressure measurement is presented in Fig. A.1 a. The granular medium is placed into a tall cylindrical container with a diameter

4:7 cm and a height 53 cm. The bottom of the cylinder is a precisely fitted, freely moving piston. The force acting on the piston is the measured by an electronic balance. Throughout the experiment the students gradually fill the cylinder with some granular material and measure the variation of the force at the bottom of the column. Contrary to the hydrostatic pressure in fluids, the pressure on the bottom of the column saturates exponentially. This phenomenon is due to frictional forces at the walls and

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a

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b 00000000000000 11111111111111 00000000000000 11111111111111 11111111111111 00000000000000 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 11111111111111 00000000000000

30 kg

Aluminum cylinder

11111 00000 00000 11111 00000 11111 00000 11111

000000000000000000000000 111111111111111111111111 111111111111111111111111 000000000000000000000000 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 granular mat. 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 111111111111111111111111 000000000000000000000000 000000000000000000000000 111111111111111111111111

paper under carbon paper

electronic balance

Figure A.1: Sketch of the experimental setups. (a) Measurement of the pressure at the bottom of a granular column. (b) Measurement of the distribution of the forces acting on single grains by the “carbon paper method” (Liu et al., 1995). inside the medium as described by the model of Janssen (1895). This model predicts:

P (z ) = g [1 e

z= ];

(A.1)

where z is the height of the column, is the density of the material, g is the gravitational

acceleration, is a constant. The students measure the saturation of the pressure, compare their data to (A.1) and determine the parameters of the model. They also analyse the fluctuations in the pressure due to the successive collapse of internal arches and the differences between independent measurements originating from the metastable nature of the investigated material. The other experiment investigates the distribution of the forces acting on single grains (P (f )). The experimental apparatus is presented in Fig. A.1 b. The granular material is confined in a container and a large force is exerted on its top surface. A carbon paper is placed under the sample, thus the grains leave marks that is proportional to the force acting on them. The relationship between the mark size and the acting force is also determined experimentally. The propagation of the forces on the random network of the grains leads to an exponential distribution of the forces. The simplest interpretation of this phenomenon is 103

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given by the q-model of Liu et al. (1995, for details see Coppersmith et al., 1996), which predicts:

P (f ) / f e

f=f0 ;

(A.2)

where and f0 are constants. The exponential tail of the distribution implies that the weight of large forces is much larger than in the case of a Gaussian distribution, thus it provides indirect evidence for the existence of force chains.

104

Appendix B Image Processing Techniques One of the most challenging tasks in our experiments consisted in extracting physical information from digitized images or videos. Since image processing techniques does not form a part of the regular studies of physicist students in our university, we think it important to share some of our experiences.

B.1 Determining the Surface Profile A great part of our measurements using the rotating drum apparatus investigated the evolution of the surface profile of the material. It was therefore crucial that the computer could determine quickly and reliably the position of the surface from the digitized images. We first note that due to the large size of image files it is essential to process the

images on-line rather than record them to hard disk. Our images consisted of 576 768 pixels, and in our typical runs we processed more than 20,000 images, leading to approximately 10 GigaBytes of information. Naturally this can be decreased considerably by compression techniques, but if we insist on lossless compression, the total size of the files still remains huge. On the other hand the extracted data consists of less than 1000 pairs of coordinates, and we store all numbers in 2 bytes thus we need only 4 kiloBytes for a given surface, and less than 100 MegaBytes for a full run. A second note is that before processing the images we have to determine the relationship

(x ; y ) = f (i ; j )

(B.1)

between the coordinates of the pixels of the image (i ; j ) and the real-world coordinates of the corresponding point of the experimental setup (x ; y ). In our case it meant that we have to determine the position of the center of the drum, the radius of the drum and the 105

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horizontal direction in our images. For this purpose we used a semi-automatic algorithm, which allowed us to specify some key points via the mouse in snapshots of the drum and a spirit level, and then calculated the (B.1) relationship automatically. The principle of the detection of the surface is that it represents a boundary between a bright and a dark region. The pixels of our greyscale images are integer values B i;j

between 0 and 255 representing brightness values between black (Bi;j = 0) and white (Bi;j = 255). Thus the fact that a region is darker means that the pixel values there are smaller. The distinction between regions of different brightness can be enhanced by the so called Look Up Table (LUT) transformations, which are simple functions that assign new brightness values to the pixels as a function of their current brightness:

0 = f (B ): Bi;j LUT i;j

(B.2)

In our setup we used a LUT transformation to make the beads sticking to the surface invisible. We manually determined the typical brightness value corresponding to the monolayer of the beads on the walls (Bth ). Then we applied the following transformation (usually called “thresholding”):

0 = Bi;j

Bi;j if Bi;j < Bth 255 if Bi;j Bth :

(B.3)

This transformation greatly enhanced contrast at the material surface (see Fig. 5.2 b and c). In order to detect a boundary between a bright and a dark region we introduce the notion of directed local contrast. Let B n be neighbouring pixels along a straight line l of arbitrary direction. Then let the directed local contrast at pixel n 0 in the direction of the line be the difference between the average brightness of the pixels in the two side of the given pixels: !

1 Cl;n0 = N

nX 0 1

n=n0 N

Bn

n0X +N

1

n=n0 +1

Bn

;

(B.4)

where N is the averaging length (we used N = 10). If then we consider a line that is (nearly) perpendicular to the surface, then the intersection of the line and the surface will be given by the point of maximum directed local contrast along the line. There are a few other questions we have to address. First, we have to decide a good method how to choose the lines along which we determine the directed local contrast (the lines should be approximately perpendicular to the surface, the position of which we

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Figure B.1: Illustration of the algorithm finding the surface profile: the first point is searched at the center of the drum (along the vertical arrow), then the algorithm proceeds along the surface (curved arrow). The next point is determined along the thin line in the slanted rectangle (the angle of this line dynamically adapts to the surface). The threshold is applied only for the pixels along this line. After reaching the wall the algorithm jumps back to the center and proceeds to the left. don’t know in advance). If the surfaces were nearly horizontal we could use only vertical lines, but since at high oil contents our surface sometimes contains nearly vertical parts, this method is not applicable. The second problem is that applying the LUT transformation on the whole image or finding the point of maximum contrast along a line spanning the whole image takes a lot of time, and our algorithm has to be fast. To solve both problems at once we utilize the information that the surface points lay next to each other, therefore if we know the position of a surface point we can find the next one in the immediate neighbourhood. Also, if we know a segment of the surface, we can also determine its approximate direction, and thus find a line that is roughly perpendicular to it. We thus used the following procedure (illustrated in Fig. B.1). We found the first surface point based on maximum contrast along the vertical line going through the center of the drum (it intersected the surface profile in all our measurements). Then we considered a short vertical segment to the right of our first point and found the second point by the same method. This way we proceeded along the surface profile until the wall. The angle 107

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of the lines and the direction of the steps dynamically adapted to the local angle of the surface (measured over a distance of several pixels to average out noise). We then turned back to the first point and found the other half of the surface. We applied the thresholding only to the pixels that were involved in the calculations of the contrast. This procedure allowed us to digitize and analyse 12 images per second with almost negligible error. This was sufficient for detecting if the flow is avalanching or continuous which was needed in measurements determining the phase diagram (see Fig. 5.8). However, the investigation of avalanche dynamics required higher resolution. Thus we developed an algorithm which recorded only the avalanches. The program digitized about 100 images at a rate of 30 frame/sec, then analysed every 10th to determine whether an avalanche has taken place. If this time interwall contained only solid-body rotation, then the images were discarded. If however an avalanche has happened then all the images were analysed and the avalanche data was recorded to disk. This method method does not guarantee that all the avalanches are recorded, but it filters out useless data, which enabled us to record the data of several hundred avalanches in a reasonable disk space.

B.2 Determining the Velocity Fields Another class of measurements concerned the velocity of the flow. Standard video recording cannot access the velocities in the bulk of the material, but they are suitable to determine velocities on the surface. Rajchenbach (1990) used a fast camera to determined the velocities from the images of single grains. This method was inaccessible for us for, because our resolution was not good enough to distinguish individual beads. Furthermore the detection of beads is very hard in our case, since we have a 3d packing of overlapping transparent balls. We used two other methods as described below.

B.2.1 Particle Imaging Velocimetry For determining velocities on the surface of the material we utilized the existence of travelling surface features. Since in the correlated and especially in the viscoplastic regime there are lasting contacts in the flowing region, the group velocity of the features is equal to the velocity of the particles (which is not necessarily true for the front velocities in the granular regime).

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Since some of the surface features are visible from the side, the flow velocity can be estimated from only the data of the surface profiles as we did it in section 5.6.1 (page 92). This however averages over velocities in the z direction. In order to determine the whole v (x; z ) field on the top surface we changed the position of the camera so that the whole surface could be recorded. The method we used was a simple “home made” algorithm based on the fundamental principle of particle imaging velocimetry. We used two successive snapshots of the flow. In the first one we chose an arbitrary region of material. We assume that during the short time interval between the two snapshots this region did not change considerably, it was only translated (in case of lasting surface features this assumption is justified, for collisional flows the method can only be used if the resolution is sufficient to distinguish individual grains). Therefore we compare the the chosen region to various regions in the vicinity of the original one (“pattern matching”), and find the most similar one. For measuring the similarity of two regions we can use for example the correlation of their pixels. Let us subtract 128 from the pixel values so that their average value is zero, and let ai and bi be the pixel values for the two regions, A and B respectively. Then the correlation of the two regions is defined as:

c(A; B ) = this way c(A; B )

P ai bi pP i P

2 i ai

2: b i i

(B.5)

= 1 if the two regions are identical, otherwise c(A; B ) < 1. We may

also allow regions that are translated by a fraction of a pixel, or rotated and calculate the pixel values by linear interpolation. In such a way sub-pixel accuracies can be reached. In our program we used a commercial pattern matching routine (provided with the Matrox Inspector package). A sample velocity field obtained by this method is shown in Fig. 5.23 (page 98).

B.2.2 Following Tracer Particles For determining the velocity field in the vertical plane we applied a completely different method, based on tracer particles. We added some larger, pink, plastic grains to the sample which were easy to distinguish from the other particles, and were visible even if they were behind a few layers of glass beads. In these measurements we digitized colour images, and used the ratio of the red and green colour band (ri = Ri =Gi ) to determine the position of the tracers. We first determined the pixels where r i was larger than a 109

A PPENDIX B.


threshold, then used a clustering algorithm to determine which pixels correspond to the same tracer particle. Clusters below a given size were discarded. Unfortunately we could not follow a selected tracer all along its path, since all tracers stayed long periods in the bulk of the material where they were invisible. Therefore we used an alternative method. We determined the position of the tracers in successive images. Then we had to determine which tracers correspond to each other in the two images. Since the time lapse between the two images was small (0:2 s), the tracers moved only small distances, we could assume that each particle in the first image corresponds to the closest one in the second. If there was any source of ambiguity, e.g. the number of tracers did not match (a tracer has disappeared) or a tracer was the closest one to two others, we discarded the images. We recorded all the measured displacements, and later we performed averaging over the displacement vectors obtained in a given region. By suitably long measurements we could obtain high-resolution velocity fields. The major problem of that method was that due to an interesting segregation effect the large tracers did not enter the central region of the drum. We plan to perform measurements with other types of tracers in the near future.

110

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Summary Well controlled experiments on idealized granular samples have been performed in order to reveal the fundamental properties of the investigated systems and facilitate future theoretical descriptions. Our studies were focused on the formation of jammed states due to the interlocking of grains, the dynamics of their intermittent collapse in the presence of slow external driving and the properties of continuous flows at high velocities. In particular three different dynamical systems were analysed. We investigated force fluctuations experienced by a object moving through a granular medium, measured the increased stability of granular samples due to an interstitial liquid, and analysed the effect of wetness on the avalanche and flow dynamics in a rotating drum. We have revealed the following interesting results: [i] It has been demonstrated that stick-slip processes are ubiquitous in granular materials, and the stick-slip model can describe both drag force fluctuations and avalanches in a rotating drum. [ii] [a] We have revealed that the stick-slip fluctuations of the granular drag force are periodic at small depth but become stepped at large depths, which can be interpreted as a consequence of the long-range nature of the force chains. [b] Both the mean force and the fluctuations were found to be almost independent of the shape and smoothness of the contact surface between the grains and the dragged object, implying that the drag force originates in the bulk of the material. [iii] [a] Our repose angle measurements via the draining crater method demonstrated that the gradual increase of liquid content leads to three fundamentally different regimes of behaviour: the granular, the correlated and the viscoplastic regime. [b] We have compared our data to two possible theoretical descriptions and determined their region of validity. [iv] [a] We have shown that the three wetness regimes are manifested in the dynamics of wet granular materials in a rotating drum, and mapped out the phase diagram of dynamical behaviour. [b] Detailed data has been presented on the dynamics of avalanches. These reveal that the increasing liquid content leads to a change in behaviour from propagating avalanche fronts to slides along internal slip planes and then to coherent motion of the surface. Furthermore, we have found [c] logarithmic strengthening at the resting state, [d] large avalanche size fluctuations at intermediate liquid contents, and [e] a novel type of pattern formation in the wettest samples. [v] The viscoplastic continuous flow has been investigated in detail. The measurements indicated that [a] there are lasting contacts in the flowing layer, [b] the flow depth is approximately independent on the flow rate, and [c] our limited data on the velocity field of the flow can be reproduced by the Bingham fluid model. 126

¨ Osszefoglal´ o Idealizált k´ısérleteket hajtottunk végre szemcsés anyagokon hogy felfedjük a vizsgált rendszerek alapvet˝o tulajdonságait e´ s adatokat szolgáltassunk mikroszkopikus modellek feláll´ıtásához. A vizsgálataink középpontjában az o¨ sszetorlódott a´ llapotok kialakulása, lassú küls˝o behatások következtében való szakaszonkénti a´ trendez˝odése e´ s a nagyobb sebességek esetén fellép˝o folyamatos folyás tulajdonságai a´ lltak. Három különböz˝o dinamikai rendszert vizsgáltunk. Jellemeztük a szemcsés anyagban el˝orehaladó szilárd testre ható fékez˝oer˝o fluktuációit, megmértük az anyaghoz adott folyadék hatására bekövetkez˝o stabilitásnövekedést, e´ s le´ırtuk a nedvesség hatását egy forgódobban fellép˝o lavinák e´ s felsz´ıni folyás dinamikájara. A következ˝o e´ rdekes megfigyeléseket tettük: [i] Megmutattuk, hogy a tapadó-csúszó folyamatok igen gyakoriak szemcsés anyagokban, e´ s a kapcsolódó modellek sikerrel alkalmazhatók mind a szemcsés anyagban fellép˝o fékez˝oer˝o, mind a forgódobban lezajló lavinák le´ırására. [ii] [a] Feltártuk, hogy a szemcsés anyagokban fellép˝o fékez˝oer˝o kis mélységek esetén periodikus, de nagyobb mélys´gek esetén lépcs˝ozetessé válik. Az a´ talakulást hosszútávú er˝oláncok jelenlétével magyaráztuk. [b] Megmutattuk, hogy mind az a´ tlagos er˝o, mind a fluktuációk csaknem függetlenek a szemcsék e´ s az a´ thatoló szilárd test közötti felület simaságától e´ s alakjától, ami arra utal, hogy az er˝o nagyságát az anyag belsejében lezajló folyamatok határozzák meg. [iii] [a] Az anyag nyugalmi szögére vonatkozó méréseink szerint a nedvesség fokozatos növelése három lényegesen különböz˝o viselkedéstartomány kialakulásához vezet: ezeket granuláris, korrelált e´ s viszkoplasztikus tartománynak neveztük el. [b] Adatainkat o¨ sszevetettük két különböz˝o elméleti le´ırással e´ s meghatároztuk azok e´ rvényességi körét. [iv] [a] Megmutattuk, hogy a három nedvességtartomány közti különbségek a forgódobba helyezett nedves minták dinamikájában is megmutatkoznak, kimértük a dinamikai viselkedés fázisdiagramját. [b] Részletesen vizsgáltuk a lavinák lefolyását. Feltártuk, hogy a növekv˝o folyadéktartalom hatására a viselkedés gyökeresen megváltozik, a felsz´ınen terjed˝o frontok helyett el˝obb az anyag belsejében lezajló csúszás, majd az egész felsz´ın koherens folyása lép fel. Megfigyeltük továbbá: [c] nyugalmi a´ llapotban az anyag logaritmikus er˝osödését, [d] közepes olajtartalmú mintákban a lavinák méretének er˝os fluktuációját, valamint [e] egy u´ jfajta mintázatképz˝odési jelenséget. [v] Részletesen megvizsgáltuk a folytonos viszkoplasztikus folyás tulajdonságait. A k´ısérleteink tanúsága szerint [a] a részecskék tartósan kapcsolódnak a szomszédaikhoz a folyó rétegen belül, [b] a folyás mélysége megközel´ıt˝oleg független a folyás sebességét˝ol, e´ s [c] a sebességeloszlásra vonatkozó korlátozott mennyiség˝u adatunk e´ rtelmezhet˝o a Bingham folyadékok elméletével.

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