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Stick-Slip in Powder Flow: A Quest for Coherence Length

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STICK-SLIP IN POWDER FLOW: A QUEST FOR COHERENCE LENGTH By MARINUS JACOBUS VERWIJS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Marinus Jacobus Verwijs

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This document is dedicated to professor Brian Scarlett.

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iv ACKNOWLEDGMENTS I would like to acknowledge my advisors Dr. Kerry Johanson and Dr. Spyros Svoronos, for their guidance and support. Dr. Johanson’s extensive experience and knowledge in powder technology were invaluab le sources of information. Dr. Svoronos’ outside view and critical approach were ve ry useful and his help in administrative manners was indispensable. My special grat itude goes to professor Brian Scarlett who always showed his trust in me. I thank hi m for convincing me to pursuit a doctoral study and to encourage me to make my own path. Thanks are also due to all the other faculty members who helped me with discussions and suggestions. I thank my committee members Dr. Anuj Chauhan, Dr. Jennifer Curtis, and Dr. Ray Bucklin for their helpful suggestions. I enjoyed the collaboration with Dr. Bucklin in the developm ent of a new, large size, Jenike Tester. I would like to thank Dr Brij Moudgil and the Na tional Science Foundation’s Engineering Research Center for Particle Science and Technology and our industrial partners for their financial support. I would also like to thank the present and past administration and staff of the Particle Engine ering Research Center for their help during my stay. I would like to gratefully acknowledge my group members including Dr. Nicolaie Cristescu, Dr. Yakov Rabinovich, Dr. Olesya Zhupanska, Dr. Claudia Genovese, Dr. Ali Abdel-Hadi, Dr. Ecivit Bilgili, Dr. Mario Hubert, Dr. Dimitri Eskin, Dr. Nishanth Gopinathan, Dr. Madhavan Esayanur, Dr. Cane r Yurteri, Dauntel Specht, Rhye Hamey,

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v Maria Palazuelos Jorganes, Stephen Tedesc hi, Milorad Djomlija, Osama Saada, Julio Castro, Benjamin James, Mark Pepple, and B ill Ketterhagen for their help in carrying out my research, and also for their support and encouragement. I would like to thank Scott Brown for his help in several matters. Many thanks go to the undergraduate students who helped me with my research, especial ly Aaron Gfeller and Jesse Schrader. I thank my parents for always believing in me and trusting me. They have always been there for me when I needed them. I apologize that I do not pursuit my endeavors closer to their house so I can see them more often. My time in Gainesville would not have been as enjoyable without all my friends at the PERC, Team Florida, and the Trigators a nd I thank them for th at. I thank Rhye for being a great roommate for four years. Lastly but mostly I want to thank my girlfriend Dauntel for being so patient with me.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF FIGURES...........................................................................................................ix ABSTRACT.....................................................................................................................xi v CHAPTER 1 INTRODUCTION........................................................................................................1 Solids Handling Problems in Industry..........................................................................2 Scientific Approach to Powder Technology.................................................................4 Stick-slip in Powder Flow............................................................................................6 Focus of Dissertation....................................................................................................7 Outline of Dissertation..................................................................................................8 2 POWDER TECHNOLOGY.......................................................................................10 Powder Flow...............................................................................................................10 Macroscopic Scale......................................................................................................12 Powder Mechanics...............................................................................................12 Constitutive Models.............................................................................................16 Powder Structure.................................................................................................19 Powder Testers....................................................................................................20 Direct Shear Cell Measurements.........................................................................22 Microscopic Scale.......................................................................................................24 Pressure Mapping................................................................................................24 Inter-Particle Forces............................................................................................25 Contact Mechanics..............................................................................................29 Discrete Element Modeling.................................................................................33 Fabric Tensors.....................................................................................................34 3 INHOMOGENEITY AND ANISOTROPY IN SHEAR TESTERS.........................37 Effective Yield Locus.................................................................................................37 Flow Function.............................................................................................................43 Influence of Moisture..........................................................................................43 Directional Dependence......................................................................................44

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vii Angle of Internal Friction....................................................................................46 Normal Stress Distribution..................................................................................48 Geometric Anisotropy.........................................................................................49 Modeling of Anisotropy in Biaxial Shear...................................................................52 Biaxial Experiments............................................................................................52 Set-up Biaxial Simulation....................................................................................54 Stress and Porosity Distributions.........................................................................55 Microstructure.....................................................................................................58 4 UNIAXIAL COMPACTION OF POWDERS...........................................................61 Design of the Uniaxial Tester.....................................................................................62 Uniaxial Experiments.................................................................................................65 Uniaxial Compaction of Mi crocrystalline Cellulose...........................................66 Uniaxial Compaction of Polystyrene Powder.....................................................67 Modeling of Uniaxial Compaction.............................................................................69 Influence of Cell Geometry.................................................................................70 Pseudo-Stress Control versus Strain Control......................................................74 5 STICK-SLIP IN POWDER FLOW............................................................................76 Introduction to Stick-Slip...........................................................................................77 Experimental Setup.....................................................................................................82 Results of Schulze Tests.............................................................................................83 Influence of Normal Stress..................................................................................87 Influence of Shear Velocity.................................................................................89 Influence of Particle Size.....................................................................................91 Influence of Moisture Content.............................................................................93 Results of Uniaxial Tests............................................................................................96 Mechanism of Stick-slip.............................................................................................98 6 INTER-PARTICLE FORCE MEASUREMENTS...................................................101 Atomic Force Microscope........................................................................................102 Experimental Setup...........................................................................................109 Adhesion Force Measurement...........................................................................110 Friction Force Measurement..............................................................................111 Inter-particle Force Modeling...................................................................................116 Atomic Force Measurement Simulation............................................................117 Friction Modeling..............................................................................................119 Stick-slip Modeling...........................................................................................123 7 CONCLUSIONS AND SUGGESTI ONS FOR FUTURE WORK..........................129 Conclusions...............................................................................................................130 Suggestions for Future Work....................................................................................134

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viii APPENDIX A CALIBRATION SCHULZE TESTER.....................................................................136 B PARTICLE SIZE DISTRIBUTION POLYSTYRENE...........................................137 C STIFFNESS CALCULATION SCHULZE TESTER..............................................138 LIST OF REFERENCES.................................................................................................140 BIOGRAPHICAL SKETCH...........................................................................................146

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ix LIST OF FIGURES Figure page 1-1 Comparison of planned and actual st artup time for solids handling plants...............3 1-2 Average startup time (left) a nd performance (right) of plants...................................4 1-3 Pyramid of knowledge...............................................................................................5 2-1 Hvorslev diagram.....................................................................................................11 2-2 Family of yield loci..................................................................................................12 2-3 Stresses on a powder sample with the corresponding stress tensor.........................14 2-4 Schematic of deformation with corresponding strain tensor....................................16 2-5 Possible shapes of the failure surface ; hexagonal (left) and triangular (right).........18 2-6 Operating window for stress control of a biaxial (right) and von Karman tester.....21 2-7 Schematic of the Schulze tester................................................................................22 2-9 Schematic of a Tekscan sensor pad..........................................................................25 2-10 Force-distance relationships for th e Born repulsion, London van der Waals attraction, electrostatic repulsion, and combinations of those..................................28 2-11 Schematic representation of a liquid br idge between a sphere and a rough plate....29 2-12 Schematic representation of Hertzian and JKR contact radius and deformation.....31 2-13 Angular distribution of parameter ; near isotropic (left) and anisotropic (right)..35 3-1 Critical Mohr circles with tangent effective yield locus for BCR Limestone, Measured with a Schulze Cell..................................................................................38 3-2 Effective yield loci of 40 m polysty rene powder for different powder moisture contents, measured with a Schulze cell....................................................................39 3-3 Schematic representation of the critical strength ( fc *) of a powder..........................40

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x 3-4 Method of computing critical strength from flow function......................................41 3-5 Effective angle of internal friction according to Jenike...........................................42 3-6 Flow function of polystyrene powde r for different moisture contents.....................44 3-7 Comparison of strength measurements of BCR limestone with the Schulze cell (PERC) with Akers and Saraber et al.......................................................................45 3-8 Flow function of BCR Limestone for a standard and reverse experiment...............46 3-9 The angle of internal friction () as a function of the majo r principal stress for a standard and reverse experiment..............................................................................47 3-10 Typical normal stress profile at the bottom of a slice of the Schulze cell................48 3-11 Normal force on the bottom of a Schul ze cell during steady state deformation of BCR limestone.........................................................................................................49 3-13 Stress profile at the bottom of a Jenike cell during pre-shear and failure in the forward direction, us ing silica powder.....................................................................51 3-13 Stress profile at the bottom of a Jeni ke cell during pre-shear in the forward direction and failure in the backward direction........................................................51 3-14 Typical result of a standard experiment with the Flexible Wall Biaxial Tester.......53 3-15 Typical result of an an isotropic experiment with the Flexible Wall Biaxial Tester........................................................................................................................5 3 3-16 The direction of wall movement at differe nt stages of the biaxial test simulation..55 3-17 Normal stresses and porosity distri butions after biaxia l consolidation....................55 3-18 Normal stresses and porosity distributions after pre-shear......................................56 3-19 Normal stresses and porosity dist ributions after forward failure.............................58 3-20 Normal stresses and porosity distri butions after anisotropic failure........................58 3-21 Normal contact force intensity distribu tion after different steps of the biaxial simulation.................................................................................................................59 4-1 Picture of the Uniaxial Tester...................................................................................62 4-2 Top view picture of the Uniaxial Tester...................................................................63 4-3 Different sample holder box designs........................................................................64

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xi 4-4 Uniaxial compaction of Mi crocrystalline Cellulose PH101....................................67 4-5 Stress controlled uniaxial comp action and relaxation curve of 40 m polystyrene powder......................................................................................................................68 4-6 Strain controlled uniaxi al compaction curve of 40 m polystyrene powder...........68 4.6 Sample box geometries used for the simulations.....................................................70 4-7 Overall stress on the xand y-walls of two cell geomet ries during uniaxial compaction...............................................................................................................71 4-8 Distributions of normal stress in th e x-direction for geometry 1 (left) and geometry 2 (right).....................................................................................................72 4-9 Fabric tensors for uniaxial compact ion using two different geometries..................73 4-10 Consolidation curves for uniaxial simulations.........................................................75 5-1 Schematic representation of s tick-slip due to surface roughness.............................78 5-2 Schematic representation of the bifurcat ion of the friction force as a function of the shear velocity......................................................................................................79 5-3 Different types of stick-slip......................................................................................81 5-4 SEM images of cornstarch (left) and polystyrene particles (right)..........................82 5-5 Typical stick-slip si gnal from a Schulze cell...........................................................84 5-6 Picture of the potentiometer fixture to the Schulze cell lid......................................84 5-7 Typical signals for the horizontal top lid displacement (grey, left axis) and shear stress (black, right axis) during stick slip in a Schulze cell......................................86 5-8 Typical horizontal displacement of the top lid relative to the base during stickslip in a Schulze cell.................................................................................................86 5-9 Maxima, minima, and magnitude of stic k-slip as a function of normal stress for the 40 m polystyrene powder.................................................................................87 5-10 Power plot from a Fast Fourier Transfor m of a low stress experiment (left) and a high stress experiment (right)...................................................................................88 5-11 Frequency of stick-slip f as a function of the applied normal stress n for 9 m polystyrene powder..................................................................................................89 5-12 Influence of shear velocity on the stick-slip frequency of 9 m polystyrene powder measured in the Schulze cell.......................................................................90

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xii 5-13 Stick-slip length scale of 9 m polystyrene powder as a function of normal stress for three different shear velocities of the Schulze cell...................................91 5-14 Characteristic length scale of stick-slip in number of particle diameters as a function of the normal stress for th e 9 and 40 m polystyrene powder...................92 5-15 Magnitude of stick-slip as a function of the normal load for different moisture contents in 40 m polystyrene powder with linear trend lines.................................94 5-16 Slope of the trend lines in Figure 5-15 as a function of the moisture content..........95 5-17 Characteristic stick-slip length as a function of the normal stress for several moisture contents......................................................................................................96 5-18 Detail of Figure 4-6 at a low stress le vel (left) and a high stress level (right).........97 5-19 Standard deviation of stick-slip si gnal as a function of the axial stress...................97 5-20 Frequency of stick-slip signal as a function of the stress and strain........................98 6-1 Schematic of an Atomic Force Microscope...........................................................102 6-2 Schematic of lateral force measur ement with the AFM showing torsional deflection or twisting of a rectangular cantilever...................................................103 6-3 Typical AFM result for force intera ction measurements between a tip and a substrate..................................................................................................................106 6-4 Schematic showing the relative sepa ration between a cant ilever tip and the substrate..................................................................................................................107 6-5 Schematic of the cantilever, tip, and substrate, represented by springs and dashpots..................................................................................................................108 6-6 Pictures of particles attached to a cantilever (lef t and middle, 500x magnification) and the substrate of pa rticles (right, 200x magnification).............109 6-7 Typical force-displacement result for polystyrene particles..................................110 6-8 Typical AFM friction loop of a measurem ent between two polystyrene particles.112 6-9 Friction force as a function of the external normal force for polystyrene particles..................................................................................................................113 6-10 Friction force as a function of the tota l normal force for polystyrene particles.....115 6-11 Discrete element model of the AFM cantilever with attached polystyrene particle and a polystyrene substrate particle..........................................................118

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xiii 6-12 Discrete element simulation of a pull-off AFM experiment using the DMT model......................................................................................................................119 6-13 Friction loop measured with a DEM si mulation, simulating polystyrene particles with a Hertzian contact model................................................................................120 6-14 Vertical cantilever deflection duri ng a DEM simulation of a friction loop...........122 6-15 Geometry of the Jenike cell in the DEM simulation..............................................124 6-16 Shear force as a function of displ acement during a Jenike cell simulation............125 6-17 Comparison of shear experiment simulations........................................................126 6-18 Shear force as a function of the displ acement with a friction coefficient of 0.3....127 A-1 Force calibration graph for the Schu lze cell in compression and tension..............136 B-1 Particle size distributi on of polystyrene powders..................................................137 C-1 Drawing of a tension bar (top) and schematic representation for stiffness calculation (bottom)...............................................................................................138

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xiv Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STICK-SLIP IN POWDER FLOW: A QUEST FOR COHERENCE LENGTH By Marinus Jacobus Verwijs December 2005 Chair: Kerry Johanson Cochair: Spyros Svoronos Major Department: Chemical Engineering Many industries such as the food, chemi cal, agricultural, and pharmaceutical industries use powdered material, either as a product, feedstock, or intermediate. Compared with liquids and gasses, pr ocesses involving powders encompass many problems, causing reduced production and uns cheduled downtimes. Most problems occur in powder transfer processes. The problems stem from a lack of understanding of the basic principles of powders. Powder flow, which is the deformation of a bulk powder, is one method of powder transfer which is not well understood. One specific problem is stick-slip, which is a discontinuous or stepwise flow of powder Stick-slip is a known phenomenon from the field of tribology, but there is no understandi ng of stick-slip in powders. In industry, stick-slip can cause structures to vibrate vi olently, causing noise and structural damage. It also causes problems in mixing, creating serious problems in the product quality.

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xv This research investigates stick-slip a nd proposes a mechanism that causes stickslip in powders. The influence of particle size, moisture content, and stress is investigated. The experiments are conducted with a Schulze shear tester and a newly developed uniaxial tester. The performance of those testers is first established by measurements and simulations. The shear stress during stick-slip is reco rded and the magnitude and frequency of the signal are calculated. Discrete element modeling is used to simulate powder flow processes that are investigated. The results show that powder cohe sion and inter-particle friction are key factor s for stick-slip. Stick-slip events prove to have a ch aracteristic length scale, which can be correlated with the partic le size and cohesion. It is proposed that a slip event is a collapse of the structure of the powder. The structure of the powder is built up from clusters of particles. The size of these clusters defines the characteristic stick-slip length. The cluster size is a function of the cohe sion of the powder. The charac teristic length is called the coherence length. The coherence length is a measure of th e sphere of influence of individual particles into the pow der bulk. It is a parameter that can be used to distinguish between the discontinuous and continuous scale of a powder.

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1 CHAPTER 1 INTRODUCTION Powder technology is one of the oldest professions in the world. It has been developed in the fields of food preparation, ceramics, pa int production, and construction (Molerus, 1996). Some scientists believe that the great pyramids in Egypt were not built by rolling enormous building blocks from quarries to the construction site, but by forming the building blocks on-site. For th is, granular limestone was poured into a rectangular mold and hardened much like concrete. It is remarkable that such an old fiel d as powder technology is not developed up to standard. Compared with gas and liquid handling, powder technology is far behind and often still an art rather than a science. Part of the problem lies in the communication between scientists and plant operators. Me rrow (1985) linked research and development to problems in industry and found that th ere is an information breakdown between operation and basic research a nd theory building. Mo st of the problems in industry are of a mechanical or physical type rather than a chemical. Therefore, scientists in the field, who are mostly chemical engineers, believe that the problems need to be solved by the operators. So research money from govern ment or industry is not focused towards solving particle problems fundamentally. Pr oblems are often solv ed with patchwork without trying to find a fundamental or permanent solution. The stigma that powder technology does not belong to chemical engineering can be found back in the curricula of engineering sc hools. First, not many schools teach particle or powder technology to engineer ing students. When they do offer such courses, it is

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2 often taught in the civil engineering or mechanical engineering department. Few departments of chemical engineering prepar e their students for the problems concerning granular material which they will encounter during their professi onal career. Chemical engineers are the ones who design the process pl ants and often operate them as well. In a 1994 publication it is stated that this neglec t of particle technology in education is especially serious in th e USA (Ennis et al., 1994). In his publication Merrow (1985) showed that the largest mechanical problem causing plants to perform below design is fa ilure of solids transfer. There are several methods to transfer solids (i.e., particles), depending on the size and form of the solids: slurries of solids in a liquid can be pumpe d, dry powders can be conveyed on belts or in a pneumatic system, or can flow under gravit y. This dissertation ha ndles the flow of powders. It specifically looks at discontinuous powder flow, which is called stick-slip. Solids Handling Problems in Industry The study by Merrow was performed in the 1980’s and shows that plants handling solids materials perform much worse than pl ants handling gasses and liquids (Merrow, 1985, 1988). About two thirds of the plants in the study operated at less than 80% of the design and one third operated at less than 60% of the design. The average for the solids handling plants was 64% of design capacity, co mpared to a 90 to 95% industry standard. The plants also needed much longer startup times than anticipated. Figure 1-1 shows the planned and actual startup times for plants with different feedstocks. Plants with a liquid and/or gas feedstock producing solids need ed a marginally longer startup time than planned. A feedstock of refined solids, mean ing the solids have undergone some type of prior processing, show a startup time that is more than twice the planned time. When the feedstock consists of raw mate rials, e.g., minerals, this wors ens to an actual startup time

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3 of 18 months, more than three times the plan ned time. Each month a startup slips behind adds an average of $350,000 to the capital cost (in 1988). The actual cost of the delays will be higher due to capital appreciation and market loss. 0 4 8 12 16 20 Liquid-gasRefined-solidsRaw-solidsType of process plantAverage startup time (months) Planned startup time Actual startup time Figure 1-1: Comparison of pla nned and actual startup time for solids handling plants with different types of feedstock (Merrow, 1988). The study above is twenty years old and some of the data used in the study are 40 years old. After publication of the study more research money has been allocated towards particle technology in the USA. A second st udy in 2000 showed that 15 years later there had been some improvement (see Figure 1-2) (M errow, 2000). The data in the Figure are for plants starting up between 1996 and 1998. It is still apparent that plants handling solids perform worse and need more startup tim e than plants handling liquids and gasses. It should be noted that the 1985 study was for 40 plants in the USA and Canada while the 2000 study included 287 solids handli ng plants from around the world. For solids handling plants basic technical da ta such as heat and mass balances are often not available. Even if this information is available from other plants, it turns out that it is difficult to transfer this process understanding from one plant to the other. There

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4 is a lack of fundamental unde rstanding of particle and pow der behavior, which makes it difficult to extrapolate knowledge from one facility to the other. 0 2 4 6 8 1234 Type of process plantAverage startup time (months) 0 20 40 60 80 100 1234 Type of process plantOperatbility, % of design after startup Figure 1-2: Average startup time (left) and performance (right) of plants handling only solids and gasses (1), solid intermediates or products (2), refined solids feed (3), and raw solids feeds (4) (Merrow, 2000). Scientific Approach to Powder Technology The key in solving problems in particle technology lies in unde rstanding the basics. When the fundamental behavior of powders is understood, reliable process and product design is possible. In a recent paper, Leuenberg er and Lanz (2005) review the state of the art in the pharmaceutical powder technology. Th ey state what has been known for some time: Powder technology has to move from ar t to science. This is visualized in the pyramid of knowledge (see Figure 1-3). Produc t and process design, as well as process and quality control, have to move forward from a trial-anderror approach towards design based on first principles. Theref ore, research should be fo cused on the understanding of first principles. For this it is often necessary to first understand the mechanistic behavior. To work towards fundamental understand ing Professor Scarlett initiated and promoted the 4M business for particle tec hnology (Scarlett, 2002). The four M’s consist

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5 of Making, Measuring, Modeling, and Manipul ating. Measuring and modeling are tools to design and control products a nd processes, which we want to make and manipulate. In industry the focus is on making products or equipment to handle these products. Manipulation of these products and processes can optimize them. The tools to do this are measurements and models. As said, these models should be built upon first principles. The input data for the models have to be m easured with reliable te sters. Therefore, the development of models and measurements needs to go hand in hand; better measurements make better models and better models make better measurements. DATA DERIVED FROM TRIAL-N-ERROR EXPERIMENTATION DECISIONS BASED ON UNIVARIATE APPROACH CAUSAL LINKS PREDICT PERFORMANCEMECHANISTIC UNDERSTANDING1st Principles DATA DERIVED FROM TRIAL-N-ERROR EXPERIMENTATION DECISIONS BASED ON UNIVARIATE APPROACH CAUSAL LINKS PREDICT PERFORMANCEMECHANISTIC UNDERSTANDING1st Principles Figure 1-3: Pyramid of knowledge (Leuenberger and Lanz, 2005). To be able to use the 4M business properl y, a systems approach needs to be used. Every system has its own time and length sc ale. From bottom up we can for example recognize atoms, molecules, particles, powde rs, process equipment, processes, and the environment. Every system builds up the next each with its dis tinct time and length scale. To understand the first pr inciples of a system, the first principles of the system it is built from need to be understood.

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6 In powder flow the knowledge level ranges fr om trial-and-error up to in some cases a mechanistic understanding. The state of the ar t in the field is mainly based on the work by Dr. Jenike (1961; 1964), which was deve loped in the 1960’s. In his work, a mechanistic model is used based on Moh r-Coulomb behavior of powders. The model uses data from shear testers to predic t powder flow behavior and design process equipment. As Merrrow’s studies show, the models used are far from perfect. Better models as well as better testers are need ed to advance towards reliable design. An example of a better tester is a True Triaxi al Tester (Verwijs et al., 2002) which would allow full control of powder deformation. Stick-slip in Powder Flow This research investigates stick-slip in powder flow. Stick-slip is a discontinuous flow of the powder, making it deform in steps. S tick-slip is also referred to as slip-stick. It might be a chicken and egg discussion, but th e author believes that the correct term is stick-slip since the stick co mes prior to the slip. A powder can not slip without first sticking, while the powder does not ha ve to slip before it sticks. Stick-slip is a well-known phenomenon in tribology (Bhushan, 2002; Bowden and Tabor, 1950). In that field it expresses itself in squeaking door hinges, car tires, or chalk on a blackboard. The mechanism of stick-slip in powder flow is not known. Stick-slip can cause problems in industry in several ways The best-known phenomenon is silo quaking or honking (Roberts and Wensrich, 2002). Th e step-wise deformation of the powder results in violent vibrations in the silo structure, which is quaking. The empty headspace of the silo works as a soundboard maki ng a very loud honking sound. The honking sound is a noise hazard while the quaking over time can cause significant structural damage.

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7 In this research the frequency and magnitude of stick-slip even ts are investigated. The frequency or time scale can be converted into a characteristi c length scale. This length scale can be used to c onnect the discrete particle s cale with the continuous powder scale. An understanding of the stick-slip behavior will give a better understanding of powder flow in general. The author proposes that stick-slip occurs in all powders, but not all powders exhibit the effects. Focus of Dissertation This dissertation focuses on stick-slip in powder flow. Powders are modeled as a continuous medium for flow purposes. Of course powders consist of individual particles. At some length scale the influence of individu al particles is not noticeable in the bulk. The border between the discrete particle scale and continuous bulk scale is not clear. It will be different for different powders. Every powder has a certain coherence length, also called sphere of influence. The coherence le ngth describes how far the influence of an individual particle reaches into the bulk. This coherence length is a key parameter in the systems approach and will connect the pa rticle scale with the powder scale. Stick-slip is a phenomenon that exhibits it self on a bulk scale while it is caused by the structure in the powder, which consists of the individual partic les. Therefore, stickslip is an indicator of the coherence lengt h. Hence the title of this dissertation is Stick-slip in Powder Flow: A Quest for Coherence Length The aim of this work is to establish the causal links betw een several operating variables and the magnitude and frequency of stick-slip events. Based on the results, a mechanistic model is proposed which explains the origin of stick-slip. The mechanism includes the formation of a structure of clusters in the powder. The mechanistic understanding of stick-slip and these clusters will help the formation of powder flow

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8 models based on first principles. In short, the roadmap of the work consist of the following objectives: Understand the artifacts of conventional testers. Measure stick-slip as a function of powder and operating variables. Measure particle properties. Correlate particle properties with macr oscopic stick-slip using simulations. Develop a stick-slip model. Outline of Dissertation The chapters is this dissertation can be r ead separately. The structure is such that they form a complete work though. The next ch apter, chapter 2, is an introduction to the subject which is useful for all the consecutive chapters. The chapter describes the two system levels that are discussed in this wo rk, the micro scale and the macro scale, and how these two scales can be connected. The micro scale section e xplains how particles interact. The macro scale section deals w ith bulk powder properties. The traditional approach to powder flow is explained and a description is given of the commercial powder tester that is used in this researc h. Finally, tools are give n to connect the micro and macro scale. The main focus here is on discrete element method (DEM) modeling. This technique models a large number of part icles, which enables the connection between the particles, the structure they fo rm, and the macroscopic response. For reliable data a good understanding of th e test equipment is important. None of the existing powder testers is perfect, so it is important to know what the strong and weak points of different testers are. In chapter 3 some of the artifacts of commercial powder testers are investigated. Stress distributions and anisotropy in testers are measured and DEM modeling is used to expl ain the measured artifacts.

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9 Chapter 4 describes a newly developed powder tester, the uniaxial tester. The tester has the unique capability that it can deform a powder both stress controlled or strain controlled. It is often claimed that there is no difference between those two control systems, but this chapter shows that in the case of stick-slip it does make a difference. The main chapter of this dissertation is chapter 5, which describes the stick-slip research. A literature review is given of the work that is performed by others investigating stick-slip in powders. Expe rimental work is presented showing the investigation of different powder and operating paramete rs. Based on the results a mechanism for stick-slip is proposed. In chapter 6 the inter-particle forces betw een polystyrene particles are investigated with the atomic force microscope. The adhe sion and friction between the particles are measured and the results are si mulated with the discrete el ement method to validate them. Using the correct particle inte ractions, a small size shear box is simulated to capture the stick-slip behavior with th e discrete element code. The last chapter discusses the findings of the combined chapters to come to the final conclusions. This dissertation is a sma ll step on the road to powder flow modeling based on first principles. In th is chapter suggestions are ma de for future research to further advance towards the fina l goal of reliable powder flow.

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10 CHAPTER 2 POWDER TECHNOLOGY Powder Flow Powder mechanics is also referred to as powderor granular fl ow. This term might be confusing since it is differe nt from the flow of fluids. Powders can transfer shearing stresses under static conditions and have a static angle of fr iction greater than zero. This enables powders to form a heap, while fluids level off. Many powders also posses a cohesive strength after consolida tion. This allows powders to form stable structures like arches, which fluids cannot. The flow of powde rs is a deformation of the powder, which can be the collapse of a stable arch. This flow is also called failure of a powder. Before a powder fails, it dilates, which is a volume expansion. The stress level at which this dilation occurs is called th e compaction/dilation boundary. The shear stresses during slow flow or deformation of powders are usually independent of the shear rate and dependent on the mean pressure. This is the opposite in most liquids, where the shearing stresses ar e dependent on the shear rate and not on the mean pressure. Therefore attempts to describe granular flow with fluid mechanics have not been successful (Jenike, 1964). The first investigations on granular flow go back to the 18th century. Coulomb (1773) hypothesized that soils fail along a rupt ure plane. Roberts ( 1882) reported that the weight of granular material measured on the bottom of a bin reduced with increasing material head. This is due to the walls, whic h support part of the weight. Janssen (1895) showed this mathematically by using a con tinuum approach. Reynolds (1885) found that

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11 a compacted material expands before it fail s, which is known as dilantancy. Hvorslev (1937) combined Coulombs and Reynolds resu lts in a three dimensional stress-strainporosity diagram. This diagram was further developed by Roscoe et al. (1958) and is shown in Figure 2-1. The diagram shows the failure surface of granular material as a function of the normal and shear stress and the porosity of the material. YL Figure 2-1: Hvorslev diagram showing the e ffective yield locus (EYL) and a yield locus (YL) which are the projection of the cr itical state line (C 1C2) and a failure surface respectively. The space is spanned by the normal stress ( ), the shear stress ( ), and the porosity ( ). The Hvorslev surface is divided into two regimes by the critical state line (C1C2). At the left side the material fails under shear while at the right side the material consolidates. The three dimensional surface can be projected onto the two dimensional shear stressnormal stress space. The critical state line projects as the effective yield locus and the yield surf ace projects as a series of yield loci, each corresponding to some initial consolidation (see Figur e 2-2). Jenike (1961) develo ped an engineering approach

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12 to design silos based on powder flow characteri stics. Jenike’s method is still being used to design silos. c Figure 2-2: Family of yiel d loci, with the cohesion c and angle of friction given for the top locus. Macroscopic Scale Powder Mechanics The flow behavior of powders is very simila r to that of soils. Since much of the soil mechanics research advances that of powder mechanics, much can be learnt from civil engineers. The tester Jenike developed or iginates from the shear box used in soil mechanics. There are some major differe nces though. The stresses exerted on soils are usually one to three orders of magnitude highe r than the stresses powders experience. The deformation of soils is usually not as large as the deformation of a powder. Soils are often saturated with an interstitial fluid, wh ile powders are generally dry. The scope of the research in soil mechanics differs from that in powder technology. In soil mechanics one does not want flow to occur (since that might mean the collapse of a dike) while in powder technology flow is gene rally wanted. Civil engineer s are more concerned about how granular material approaches failure, i.e., the region before the compaction/dilation

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13 limit. In powder mechanics the interest is the region between the compaction/dilation limit and failure. Those factors prevent a dir ect comparison between soil mechanics and powder mechanics. In powder mechanics a continuum approach is used to describe the state of a powder. This means that the particulate stru cture and voids are disr egarded. If a sample volume is taken infinitesimally small, it will still behave similarly as the bulk and its properties are described by con tinuous functions. It allows the definition of stress at a point. For the continuity prin ciple to hold, the sample size of powders measured should be larger than the minimum sample size. Th e coherence length, i.e ., the length in which the effect of individual partic les vanishes, also describes this minimum sample size. It is the smallest volume that still behaves as a bulk powder. The minimum powder sample is generally not known and is inve stigated is this project. Besides the continuity a ssumption, two other assumptions are often made in continuum mechanics, homogeneity and isot ropy. Homogeneity means that the material has identical properties at al l points in the material. Isotropy means that the material properties are the same in all directions. In powder mechanics those assumptions can often not be made. Powder de nsity can for example vary with the bed height, which means that the sample is not homogeneous. Afte r deposition of flaky particles the ability to sustain stress in the vertical direction can be different from that in the horizontal directions, which means the sample is not isotropic. Stress (ij) is defined as a force ( Fj) in direction j that acts on an area ( Ai) with normal vector i as shown in equation 2-1. A dist inction can be made between normal

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14 stresses (ii), where the force works on the plane i in the direction i and shear stresses (ij or ij), where the force works on the plane i in the direction j i j A ijA Fi0lim 2-1 The equilibrium stresses on a cubical powde r sample can be described by a stress tensor. This is shown in Figur e 2-3, where the normal stress es are on the diagonal and the shear stresses fill the rest of the stress tensor. The tensor can be resolved such that all the shear stresses are zero. The resulting normal st resses are called the pr incipal stresses. In order of increasing magnitude, these stress es are called the minor, intermediate, and major principal stress and denoted by 3, 2, 1, respectively. In powder mechanics compressive stresses are positive and tensile stresses are negative. 33 32 31 23 22 21 13 12 11 S 33 32 31 23 22 21 13 12 11 S 33 32 31 23 22 21 13 12 11 SS Figure 2-3: Stresses on a powder sample with the corresponding stress tensor. The three principal stresses are the eigenva lues of the stress tensor and form the real roots () of the so-called characteristic equa tion 2-2. The eigenvectors of the tensor denote the direction of the principal stresses. I1, I2, and I3 are the invariants of the equation and are shown in equa tions 2-3 to 2-5. The invarian ts are independent of the rotation of the coordinate system. 03 2 2 1 3 I I I 2-2 33 22 11 1 S tr I 2-3

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15 33 31 13 11 33 32 23 22 22 21 12 11 2det det det I 2-4 S Idet3 2-5 A form of the first invariant that is commonly used is the mean stress (m) or hydrostatic stress (H), which is the normal mean of the three principal stresses. Some models use the stress deviator or deviatoric stress tensor S ’, equation 2-6, where I is the unit matrix. The directions of the principal deviatoric stresses and principal stresses are the same. The values of the prin cipal deviatoric stresses are i m. The invariants for the deviatoric stress can be calculated similarly to the invariants of the stress. The first invariant of the deviatoric stre ss will be equal to zero. In ro ck mechanics it is believed that the deviatoric stress cau ses change of sample shape and the hydrostatic or mean stress causes volumetric change For granular material this is not totally correct. A granular material can dilate when it is sheared with a deviatoric stress. I S Sm 2-6 Similar to the stress tensor, there is also a strain tensor, describi ng the state of strain or deformation of a powder sample. There are several ways to define strain. In this research the Cauchy definition is used, which is the change in length per unit of initial length. Figure 2-4 shows a schematic repres entation of the normal strain and the shear strain. Equations 2-7 and 2-8 show the calcula tion of the strain components. It can be seen that contractive strains are positive a nd extensive strains are negative, which is common in powder mechanics. Beside the norma l and shear strains, a sample can also strain by rotation. Models often use the strain rate, which is the derivative of the strain with respect to time.

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16 l2 l1 l3 l2 11 l2 l3 l3 l1 l3 11 33 32 31 23 22 21 13 12 11 l2 l1 l3 l2 11 l2 l3l2 l2 l1 l3 l3 l2 l2 11 11 l2 l3 l3 l1 l3 11 l3 l1 l3 11 33 32 31 23 22 21 13 12 11 33 32 31 23 22 21 13 12 11 Figure 2-4: Schematic of deformati on with corresponding strain tensor. i i iil l 2-7 i j j i ijl l l l 2 1 2-8 Constitutive Models To be able to predict powder flow constitutive equations have to be developed. It is difficult though to come up with a universal equation for powders. Some powders behave plastically, others elastically, some linearly but most not. Therefore, using the continuum mechanics approach, several models have been developed. Jenike (1961) used an ideal Mohr-Coulomb model of friction to describe powder flow. The Coul omb yield criterion describes plastic deformation in the shear pl ane. The model assumes that the shear stress in a failure plane is indepe ndent of strain or strain ra te, but dependents on the normal stress on the shear plane as represented in Figu re 2-2. The yield loci can be described by equation 2-9, where is the friction coefficient, the angle of friction, and c the cohesion. The cohesion is a function of the compaction or “history” of the powder. The failure (or strength) of a powde r is therefore dependent on that history. Jenike formulated a way to measure and calculate this stre ngth, which is called the unconfined yield strength fc. The plot of this unconfin ed yield strength as a func tion of the pre-shear stress

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17 is called the flow function. This model is ma inly used in engineering to calculate the limiting state of stress in pow ders, e.g., during failure. The Mohr-Coulomb criterion can usually predict the compaction/dilation limit, but can not predict the full stress-strain curve. c c ) tan( 2-9 The von Mises model predicts that yield occurs when the value of the second deviatoric stress invariant is equal to a consta nt. This constant is a material property that indicates the onset of yield. The deviatoric stress does not include the hydrostatic stress. It is known that the compaction of the powde r does influence the strength of a powder. Therefore Drucker and Prager (1952) included the first inva riant of the stress tensor. Equation 2-10 shows the Drucker-Prager model where, I2 ’ is the second invariant of the deviatoric stress, I1 the first invariant of the stress, DP a frictional coefficient, and kDP a cohesive coefficient. DP DPk I I 1 2 2-10 The model assumes plastic deformation when the stress state of the powder is at the failure surface. Most powders deform before th at point, which is called strain hardening. Powders are better described with an elasto-pla stic strain-hardening m odel. Drucker et al. (1957) introduced a hardening cap to their model which accounts for the volumetric plastic deformation before failure. The cha nge of bulk powder density before failure (dilatancy) is used in the model. Further development of models was thr ough the Cam-clay model and the modified Cam-clay model. The Cam-clay model assume d a bullet-shaped cap, while the modified

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18 Cam-clay model assumes an elliptical shape. The elliptical model is given in equation 211, where M is a material parameter and 0 is a strain hardening parameter. M Im m 2 0 2 2-11 All models described above rely on the firs t two invariants of the stress or stress deviator. It is well known that in the failure of geomaterials the th ird invariant is also involved. This is shown in the right figure in Figure 2-5 by the conical shape of the failure surface. If the shape of the deviatoric plane (distance from the origin) would be circular, it could be described by the second invariant of the de viator and the mean stress. A point P on the surface can be found using the second and third invariant of the deviator, as shown in equation 2-12. P 0 213 ) 3 1 3 1 3 1 ( n2 1 3 ) 3 3 3 (3 2 1 P P 0 213 ) 3 1 3 1 3 1 ( n2 1 3 ) 3 3 3 (3 2 1 P Figure 2-5: Possible shapes of the failure su rface; hexagonal (left) a nd triangular (right). 2 3 3 cos' 3 2I I 2-12

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19 For many geomaterials the failure surface is not circular but triangular with rounded corners as shown in the figure. For powders the failure surface has a hexagonal shape. The corners of the hexagon might be rounded as well. To describe the roundness of the corners, the third inva riant of the deviator is needed. To calculate the third invariant the full stress tensor is needed. The full stress tensor can only be measured with a true triaxial tester. Powder Structure The current flow models assume that the powders are homogenous and isotropic. They do not take the structure of powders in consideration. Especially cohesive powders are known to be non-homogenous. They form a gglomerates and cluste rs with different porosities. The structure that is formed in this way is difficult to de scribe, especially since porosity is a scalar parame ter (Scarlett et al., 1998). The isotropic assumption is a simplification of reality as well. It has been shown that most powders have a tendency to show anisotropic behavior. Following Wong and Arthur (1985), there are two different forms of anisotropy, namely inherent and induced. Inherent anisotropy is the result of the deposi tion process and the particle characteristics. Therefore it is also called stru ctural anisotropy. It is found th at after raini ng the particles in a tester, the contact planes between particles were orient ed mostly horizontally due to the particle’s own weight (Feda, 1982). The de gree of structural anisotropy will vary with different deposition processes and particle characteristics. Part icles that are elongated or flake-like will show more anisotropy than spherical particles. Induced anisotropy, also called mechanic al anisotropy, is due to the strain associated with an applied stress. According to Li and Puri (1996), the mechanical anisotropy is the result of structural anis otropy. This is not true, because a powder

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20 existing of spherical particles can also be anisotropic (Zhupanska et al., 2003). Induced anisotropy can be significant. The effect of anisotropy is most clearly shown in the angle of internal friction and the unconfined yield strength, which can change by 10% (Arthur and Menzies, 1972) and 200% (Sarab er et al., 1991), respectively. Although a distinction is made between i nherent and induced anisotropy they are actually the same. Inherent anisotropy is induced by the deposition process, which is a mechanical process. So inherent anisot ropy is a form of induced anisotropy. Powder Testers An extensive comparison between powder testers is given by Schwedes (2003). Powder testers can be divided into direct shear testers and indirect shea r testers. In direct shear testers the location of the shear plane or zone is defined by the geometry of the tester. They are based on the shear box model and examples are the Jenike shear cell, the Peschl cell, and the Schulze cell. These test ers are able to measure the flow function, following a precisely described proc edure (ASTM, 2000, 2002; EFCE, 1989). The results from different testers vary due to the different geometries of the sample holders. The exact stresses and strains in dir ect shear testers are not known (Bilgili et al., 2004; Janssen, 2001). It is assumed that there is a uniform stress and st rain in the testers, but that is not the case. The control of direct testers is limited. In general only the normal stress, and shear strain rate can be controll ed and only the shear stress can be measured. Therefore, direct shear tester s are not suitable for fundament al research towards general constitutive laws. Indirect shear testers do not force a shear z one in the powder at a specific location, but deform the powder as a whole. The powde r will form shear zones where it needs to and this should be independent of the geometry of the tester when th e sample size is large

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21 enough. Most indirect shear testers are biaxial cells. These testers ha ve more control than the direct shear cells. In gene ral two principal stresses or strain rates can be controlled and the third principal stress is measured. An example of a biaxial tester is the Flexible Wall Biaxial Tester which is discussed in chapter 3. Civil engineers use a cylindrical tester, wh ich is called the von Karman tester. They claim that this tester is a tria xial tester. It is assumed that the hoop stress is equal to the radial stress, so those two principal stresses cannot be controlled independently. It is very difficult to use this tester for powders since the tester cannot measure in the low stress region that is important for pow der flow (Verwijs et al., 2003). It is not certain whether the whole powder sample is in deformation. To get a useful strain tensor, the whole powder sample has to be in deformation. For this, knowledge of the minimum sample length is important. The von Karman tester is not suitable for the fundamental research conducted in this work. Figure 2-6 shows the operating window for st ress control of a biaxial and the von Karman Tester. It can be se en that both testers cannot be controlled in the full three-dimensional stress regime. Ther efore a true triaxial tester is needed. IIIIII II II I I I3II1III I3II1III I1 II 3 III IIIIIIIIIIII II II I I III II II I II I I I3II1III I3II33II1III IIII11IIIIII I3II33II1III IIII11IIIIII I1II11 II 3 III II II 3 3 III III IIIII I II II I I IIIIIIIIII I I II II I I III II II I II I I Figure 2-6: Operating window for stress control of a biax ial (right) and von Karman tester (left). For soil mechanics a few true triaxial tester s exist. One is located at the university of Bristol (Airey and Wood, 1988). This tester is not suitable for powder flow research

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22 though. The stress regime the te ster is designed for is orders of magnitude higher than the stresses occurring in powder flow. The r obust design necessary to accommodate these stresses is far from frictionless. The friction in the system is negligible for the stresses applied in soil mechanics, but is higher than the stresses that are used in powder flow. Therefore, measurements in this lo w stress regime will be unreliable. Direct Shear Cell Measurements The most widely used type of powder tester is the direct shear cell. The best known are the Jenike shear tester, the Schulze shear te ster, and the Peschl tester. The tester used in this research is the Schulze tester. The te st procedure is very si milar to other direct shear testers, especially the data reduction. The Schulze tester is an angular direct shear cell that allows the determination of the unconf ined yield strength as well as the internal angle of friction and the effec tive angle of internal friction. The advantage of the Schulze tester is that it has unlimited travel. The disa dvantage is that the te ster shears the powder in an angular direction, a nd not linearly as in the Jeni ke cell. Figure 2-7 shows a schematic of the Schulze tester. Figure 2-7: Schematic of the Schulze tester.

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23 A powder sample is placed in the base (whi ch has a rough bottom) that is rotating. A lid with vanes is placed on top of the powder, such that the vanes stick into the powder. The top lid is held in place by tension bars preventing it from ro tating. The bars are connected to load cells which measure the force that is necessary to keep the top in place. Through a hanger the top lid can be loaded w ith weights to compact the powder with a normal stress. Since the bottom is rotating and the top is stationary, the powder will shear somewhere between the bottom and the vanes. The load cells measure this shear force. The standard method to measure the unconf ined yield strength of a powder is a steady-state deformation under a certain norma l load, followed by a failure with a lower load. The procedure is described in an AS TM standard (2002). For normal experiments the powder is deformed with a constant cloc kwise angular velocity until the shear stress reaches a constant value. This step is calle d the pre-shear and at this point there is no change of density and the powder is in steady state deformation. The state of stress of the powder at this point is the cri tical state. Once this state is reached for a certain normal load, the deformation is stopped and the normal load reduced. With this reduced load, the shearing is started again to fail the powder. Th is sequence is repeated several times with the same load during steady state deformation but different failure loads. In the shear stress-normal stress space thes e failure points form the yiel d locus corresponding to the chosen critical state of the powder. From the yield locus the unconfined yield locus can be found by extrapolating a Mohr circle through the origin and tangent to the yield locus (see Figure 2-8). The major principal stress of this Mohr circle is the unconfined yield stress fc. The major principal stress of the Mohr circle through the st eady state point and tangent to the yield locus is called the consolidation stress n. The combination of the

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24 consolidation stress and unconfin ed yield stress for several yiel d loci will form the flow function. Normal Stress (kPa) Shea r Stress (kPa) fc n Figure 2-8: Diagram showing th e yield locus (solid line), steady state point (diamond), failure points (triangles), major Mohr circ le (dashed semicircle), and unconfined Mohr circle (dotted semicircle). Microscopic Scale Although the focus in the powder flow fiel d has been on the macroscopic scale for many years, more and more researchers are inve stigating the microscopic particle scale to explain macroscopic phenomena. As explained in the previous chapter, the macroscopic scale is built up from the microscopic s cale. A good understanding of the processes on the microscopic scale will enable the explanation of macroscopic phenomena. Pressure Mapping It is generally known that the stresses th at are measured with conventional powder testers are average stress values. In powde r testing, the overall normal force on the powder is divided by the area the force acts on to give the normal stress acting on the top of the powder sample. This normal stress is assumed equal to the normal stress acting throughout the material. This is not correct since the walls of the tester will carry part of the load. Because of this, there will be a st ress distribution in both the radial and axial directions within the powder ra ther than one overall stress.

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25 Within the measured area there can be larg e deviations in the local stress. These deviations lie in between the microscopic a nd macroscopic scale. The stress deviations can be caused by individual particles pressing against the wall forming hot spots, which is a microscopic phenomenon. The deviations ca n also be caused by the structure of the powder, which is in between the two length scales. In this work the Tekscan mapping techni que (Hunston, 2002) is used to measure the stress distributions on surfaces. The Tekscan sensor consists of 1936 sensels (pressure sensor points) arranged in a square grid c onfiguration (see Figure 2-9). The sensor pad has a resolution of 15.5 sensels per cm2. Experiments conducted w ith these pads revealed that pressure measurements are accurate to wi thin 10% of the measured stress when used in powder shear conditions (Bilgili et al., 2004; J ohanson and Bucklin, 2004). Figure 2-9: Schematic of a Tekscan sensor pad. Inter-Particle Forces Forces of different origin can exist between particles. The most common forces are the van der Waals forces, electrostatic for ces, electromagnetic forces, and capillary

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26 forces. These forces have a physical origin and are called long-range forces. Short-range forces are interactions invol ving covalent bonds and usually act up to a distance of 0.1 – 0.2 nm. The long-range forces act beyond this point up to several nanometers. Not all long-range forces have to be present or si gnificant in a particle system. The dominant forces in this research are the van der Waals forces and capillary forces. The van der Waals forces can be subdivid ed into the London dispersion force, the Keesom force, and the Debye force. The L ondon dispersion force is an induced dipole interaction between atoms and molecules, creat ed by fluctuations of electronic charges. The Keesom attraction force is an interact ion between rotating permanent dipoles. The Debye force is an interaction between rota ting permanent dipoles and polarizable atoms and molecules. The London force is usually th e dominant force since all materials are polarizable, while for the other van der Waal s forces permanent dipoles are required. A detailed description of the van der Waals fo rces can be found in several books (Bhushan, 1999; Israelachvili, 1985). For atoms, the London dispersion potential is combined with the Born repulsion potential in equation 2-13, which is called the Lennard-Jones Potential W, with B the Born repulsion constant, C the London dispersion constant, and D the separation of the atoms. The Born repulsion is due to overlap of electron clouds of two bodies when they approach within 2 to 3 A. 6 12D C D B W 2-13 On a particle scale, when the separation distance D between two identical spheres is small compared to their radius R, the van der Waals interaction forces FvdW can be approximated with equation 2-14, where A is the Hamaker constant. The van der Waals

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27 forces between bodies that are not spherical or of dissimila r size can be calculated using their specific geometries. The result for the interaction between two spheres of different size and the interaction of a s phere with an infinite plate are given in equation 2-15 and 216, respectively. Two identical spheres: 212 D AR FvdW 2-14 Two spheres of different radius: 2 1 2 1 26 R R R R D A FvdW 2-15 Sphere with plate: 26 D AR FvdW 2-16 Derjaguin proposed a method to approxima te the interaction between surfaces based on the interaction free energy per uni t area. Using a slice method the adhesion force for different geometries can be calcula ted. The adhesion force between a sphere with radius R and a plate using Derjaguin’s approxi mation is calculated using equation 217, where W123 is the work of adhesion between tw o plates of species 1 and 3, with species 2 in between. 1232 RW Fadh 2-17 Electrostatic forces generally act across a longer range than van der Waals forces. In wet systems the surface can partially dissoc iate and form an electrostatic double layer. This double layer causes a repulsive electrostat ic force between surfaces. In dry powders dissociation of the surfac e is not very likely. Fo r non-conducting or poor-conducting materials it is possible though to accumula te surface charges. A common manner to acquire these surface charges is through pneumatic conveying of powders. The electrostatic interaction between particles can be attractive or repulsive. Figure 2-10

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28 shows a typical repulsive el ectrostatic force between tw o bodies as well as the London force and Born repulsion as a functi on of the distance between the bodies. Separation Distance (nm)Interaction Force (nN)1. Born repulsion 2. van der Waals attraction 3. Born repulsion & van der Waals attraction 4. Electrostatic repulsion 5. All interactions combined 1 5 4 3 2 Figure 2-10: Force-distance re lationships for the Born repulsion, London van der Waals attraction, electrostatic repulsion, and combinations of those. An additional force that can exist between particles is a capillary force. A liquid bridge can form between particles even when a powder is relatively dry. An extensive review of inter-particle capi llary forces is given by Esayanur (2005). In his work Esayanur connected capillary forces with powder cohesion. Rabinovich et al. (2000a; 2000b) developed a simplified relations hip for a capillary adhesion force Fad between a smooth spherical particle and a flat substrate with nanos cale roughness. The relationship is given in equation 2-18, where is the surface tension of the liquid, R the radius of the adhering particle, Hasp the maximum height of the aspe rities above the average surface plane, r the lesser radius of meniscus, and cos = (cos P + cos S)/2. The angles P and S

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29 are the contact angles of the liquid on the adhe ring particle and substrate, respectively, as shown in Figure 2-11. Equation 2-18 is appl icable only when the meniscus is large enough to span the distance between the adhe ring particle and the average surface plane and when the radius of the meniscus is small compared to the adhering particle. cos 2 1 cos 4 r H R Fasp ad 2-18 r p H asp s r H asp R Figure 2-11: Schematic representation of a liquid bridge between a sphere and a rough plate (with permission, from Esayanur (2005)). Contact Mechanics When considering a powder bed on a particle scale, besides the interaction forces, the particle contact behavior is of eminent importance. When there is no significant plastic deformation, the contact area depends on the elastic response of the material. In 1888 Hertz developed a contact mechanics theo ry between an elastic spherical body and a plate. His work forms the basis of severa l theories. In his theory Hertz assumed no inter-particle forces, so a c ontact deformation occurs due to an external compression force F When a particle and a plat e are pressed together, they form a circular contact area. The radius a of the contact area follows a rela tionship as shown in equation 2-19,

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30 where R is the particle radius and K the reduced elastic modul us. The reduced elastic modulus can be calculated from the Poisson’s ratio and the elastic modulus E of the two materials using equation 220. The contact deformation shown in Figure 2-12, can be calculated using equation 2-21. 3 1 K RF a 2-19 sphere sphere plate plateE E K2 21 1 4 3 1 2-20 R a2 2-21 In the 1970’s two famous theories were developed from Hertz’s work; DMT mechanics and JKR mechanics. The DMT theory is developed by Derjaguin, Muller and Toporov (1975) and includes long-range attractio n forces to Hertz’s theory as shown in equation 2-22, where Fadh is calculated using equation 217. The contact deformation is Hertzian, so calculated usi ng equation 2-21. The model shows no hysteresis between loading and unloading, thus excluding the possi bility of pull-off forces. The applicability of the model is limited to stiff systems with low adhesion forces. 3 1 adhF F K R a 2-22 The JKR theory is developed by Johnson, Kendall and Roberts (1971) and does not include the long-range forces outside the cont act area but does include short-range forces inside the contact area. The radius of the contact area is given in equation 2-23. The model accounts for adhesion hysteresis, forming a neck at unloading. Therefore, the

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31 deformation is not Hertzian but according to equation 2-24. The JK R model is suitable for materials with a large a dhesion force and large radius. 3 1 2 123 123 1233 6 3 R W RF W R W F K R a 2-23 K a W R a123 26 3 2 2-24 JKR neck a J KR a H ert z F R Figure 2-12: Schematic representation of Hertzian and JKR contact radius and deformation. In the 1990’s two additional theories were developed; BCP mechanics and Maugis mechanics. The BCP model is an empirical model, develope d after years of force curve studies. The Maugis theory is a complex m odel, which applies to all systems and connects the region between the DMT and JKR theory. An extensive comparison between the different contact models is given by Burnham and Kulik (1999). The friction between particles is very im portant for powder flow evaluation. When a powder is sheared, particles have to slide along each other and the friction between their surfaces will define the ease with which that is accomplished. This inter-particle friction

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32 is thus a major contributor to the shear for ce. In the macroscopic world the friction of a material is defined by the friction coefficient. The friction coefficient is the ratio of the friction force at which a body starts movi ng over the normal force pressing the bodies together. Since the friction force generally is a linear function of the normal force, the friction coefficient is a constant. The value can range from near zero for frictionless materials to up to 4 for rubber and some metals. Most common materials will have a friction coefficient between 0.1 and 0.7 though. There can be a different coefficient for static and kinetic friction, where the first is slightly larger than the latter. Friction is proportional to the contact area of the contacting bodies, as shown in equation 2-25 (Adams et al., 1987). It is not the macroscopic area that is of importance, but the actual area of contact points A Due to surface roughness the actual contact area is much smaller than the macroscopic area of two bodies. When two pa rticles are sliding, adhesive junctions are formed and work is required to rupture these junctions. The strength of these junctions is repr esented by the interface shear strength A Ffriction 2-25 The contact area will increase with increas ing normal load due to deformation of the contact points. When purely plastic contac t deformation occurs, the contact area will increase linearly with the normal load since plastic yield is the quotient of the yield force and the area. When an elastic Hertzian defo rmation of the contacts occurs, the contact area will be a non-linear functi on of the normal load. From equation 2-17 it can be seen that the contact area is proportional to the nor mal force to the power two-thirds. This is why the friction force can be expressed with the empirical equation 2-26, where k is the

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33 friction factor, F the normal force, and n the load index with a value between 2/3 and 1. When n is unity the friction factor becomes the friction coefficient as discussed before. n frictionkF F 2-26 Adams et al. (1987) combined Hertz m echanics with equation 2-25 to form equation 2-27. They separated in an intrinsic shear strength at zero load 0 and a pressure coefficient to account for the pressure dependence of Using either the DMT or JKR model, adhesion can de added to th e normal force in equation 2-26 and 2-27. F K RF Ffriction 3 2 0 2-27 Discrete Element Modeling Discrete element method (DEM) modeling is generally used to investigate the effects of microscopic mechanic al properties (rheological proper ties of particles, friction between particles and the wall) on the macr oscopic mechanical response of powder materials. The critical parameters of particle packing and structure in an assembly are the changes in contact orientations (contact normals) of particles, particle arrangement (i.e., coordination number and dist ribution of neighbors), and st ructural porosity due to specifically applied loading paths. These changes in microstructure are mostly responsible for directional anisotropic beha vior of powders. Using discrete element analysis, the evolution of a stress-strain res ponse of a powder as well as the evolution of microstructure due to different lo ading histories can be simulated. DEM was introduced by Cundall in 1971. He c onsidered particulate material as an assembly of particles, which are in direct interaction to each other. The program was developed to simulate the beha vior of rock by masses that interact through springs and

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34 dashpots. Cundall and Strack (1979) developed the program to simulate the quasi-static shear deformations of soils. DEM computa tional procedures a lternate between the application of Newton’s second law of motion on particles and their contact interaction. Contact mechanics are used to define the for ce between particles that are in contact. For every particle the resultant force is calculat ed from all its contacts and additional forces like gravitational forces. Newton’s second law is used to calculate the acceleration of the particle due to its resultant force. All part icles are allowed to displace for a very short time step using their specific velocity and acceleration. The ne xt step all contacts are reevaluated and the cycle repeats. Fabric Tensors In many cases continuum mechanics cannot de scribe powder flow behavior well. In those instances the influence of individual particles or particle ensembles is too large. DEM simulations are a good altern ative to model those situat ions. DEM can be used to define the structure in the pow der, e.g., anisotropy. The micr ostructure can be described using the concept of fabric tensors. This con cept is generally accepted in the field of soil mechanics. These tensors are related to the spat ial distribution of the particles, their sizes and orientations, and, in particular, to the distribution of the cont act normals and contact areas, as well as other geometri cal entities. This is a useful way to investigate anisotropy since it will give the magnitude of a paramete r or property as a func tion of the direction. Following Kanatani (1984) the 2D invarian t formulation of the distribution density function E ( n ) of the unit vector n is given by equation 2-28. The fabric tensors of the distribution, Jij and Jijkl, are shown in equation 2-29 and 2-30, respectively. 1 ()1... 2ijijijklijklEnJnnJnnnn 2-28

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35 ij j i ijn n J2 1 4 2-29 1 16 8ijklijklijklijklJnnnnnn 2-30 An example of a distribution is represen ted in figure 2-13 for two dimensions, where is the orientation of n measured from the vertical and can be a parameter like the stress. Figure 2-13: Angular dist ribution of parameter ; near isotropic (left) and anisotropic (right). For this 2D case, the density distribution function can be written as a function of the angle as shown in equation 2-31. The coefficients A B C and D are a function of the angle and are given in equation 2-32. Usually, the second order tensor in expression 2-29 is sufficient to describe the microstructure of the material, but for highly anisotropic microstructure the fourth order te nsor in equation 2-30 is important. 11 ()2sin21cos2 2 1 188cos442sin4... EnAC CDBA 2-31 4 3 2cos cos sin cos cos sin D B C A 2-32 The distribution of the intensity of the c ontact forces is important as well. In cohesive powders, particles te nd to form clusters. Anisotropy in macroscopic mechanical

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36 response is not due to the genera l trends in orientation of al l particles, but just to the orientation of particles that ar e at the load carrying, clusterto-cluster interfaces. The particle size distribution also has to be taken into account. A broader distribution in particle size leads to a more heterogeneous powder sample and, consequently, to more anisotropic behavior. In terms of micromechanical properties, the friction and particle shape are critical as well.

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37 CHAPTER 3 INHOMOGENEITY AND ANISOTROPY IN SHEAR TESTERS This chapter discusses some known (but often ignored) as well as some new artifacts of shear cells. The intention is not to discredit these cells but to make users aware of these phenomena in powder testers. When the existence and cause of these artifacts is understood, it makes the interp retation of shear data much more reliable. Effective Yield Locus One of the most common measurements with a direct shear cell is the effective yield locus (EYL). Therefore, it is strange that there are several definitions of this locus. The original locus as defined by Jenike (1964) is a locus that starts at the origin and ends at the largest Mohr circle (see Figure 2-2). Another proposed EYL goes through the steady state deformation points of a tests seri es, making these points th e tangent points of the locus and the largest Mohr circles. A third variation th at can be found connects the tops of the Mohr circles, creating a locus of maximum shear stresses during steady state deformation. The common factor between these different EYL is that they are extrapolated to start at the origin. Considering the projection of the critical state line of the Hvorslev diagram (see Figure 2-1), it cannot be conclude d that the intercept has to be zero since the critical state line is asymptotic. The assumption that the EYL has no intercept comes from the soils mechanics field. In this field the stresses of interest are very high, making a possible small intercept insignificant. The stress levels in powders can be such that this intercept is

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38 significant. Cohesive powders generally show critical stat e Mohr circles as shown in Figure 3-1. 0 1 2 3 4 5 012345678 (kPa) (kPa) = tan(r) + cir = 38.6 ci = 0.315 kPa r ci Figure 3-1: Critical Mo hr circles with tangent effective yield locus for BCR Limestone, Measured with a Schulze Cell. The data are from experiments that are conducted with the Sc hulze tester with BCR limestone. It can clearly be seen that the be st fit for the EYL has an intercept with the yaxis. This has been explained by Molerus (1978 ), who attributed th is intercept to an inherent adhesion between the particles. Therefore, this intercept will be called the inherent cohesion of the powder. The cohesi on between the particles is related to the actual cohesion value as measured by direct shea r testing as shown in Figure 3-2. In this figure, the EYL for polystyrene powder is s hown for different moisture contents. As discussed in chapter 2, the moisture c ontent can change the cohesion between the particles due to the formation of liquid bridges between the particles. Figure 3-2 shows

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39 that this increased cohesion between the particles causes an increase in the inherent cohesion of the powder. 0.22% water: ci = 0.0 kPa 0 5 10tau (kPa) 0.40% water: ci = 0.16 kPa 0 5 10tau (kPa) 0.60% water: ci = 0.88 kPa 0 5 10 0510152025 sigma (kPa)tau (kPa) Figure 3-2: Effective yield loci of 40 m polystyrene powder for different powder moisture contents, measured with a Schulze cell. The inherent cohesion can also be obtained by data represented in a typical flow function. This can be best seen by noting that Figures 3-1 and 3-2 show all of the termination stress state Mohr circ les for the family of yield loci that are used to generate the flow function. As stated above, these stre ss state Mohr circles are tangent to the real effective yield locus. There exis ts a Mohr circle stre ss state that is ta ngent to this real effective yield locus and is coincident with a particular unconfined stress state Mohr

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40 circle. Figure 3-3 describes th is unique condition. The inte rcept for the real effective yield locus or inherent cohesion ci can be related to this critical unconfined yield strength fc* as indicated in Equation (3-1). 1 r i C Real Effective Yield Locus Yield Locus f c * Figure 3-3: Schematic representati on of the critical strength ( fc *) of a powder. r r c if c cos sin 1 2* 3-1 This unique cohesion value is for the case where the major principal consolidation stress equals the unconfined yi eld strength. It should be not ed that this condition can not be measured by a direct shear test method and must be obtained from extrapolation of the flow function. The flow func tion represents the relationshi p between the major principal stress and the unconfined yield strength. In fact, the intersection of a line passing through the point (0,0) having a slope of 1.0 intersects the flow function at this critical yield stress fc* (see Figure 3-4). Therefore, the shear stre ss intercept on the real yield locus can be found through an analysis similar to the Jenike arching analysis. This implies the curious result that stress states during steady flow de pend on the cohesive nature of the material, implying that incipient failure conditions of a bulk material influe nce flow modes where continual deformation occurs. Such a result could explain why solids flow rates are

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41 sometimes influenced by the cohesive nature of the material. It may also explain why channeling occurs in fluidized beds with c ohesive material. In fluid beds, inherent cohesion may play a major role in the flow profile. fc Slope = 1( 1 *, fc *) Flow Function Figure 3-4: Method of computi ng critical strength from flow function to be used for inherent cohesion calculation. Since the Jenike procedure forces the EYL through the or igin, that procedure will produce a different effective angle of internal friction for every stress state as shown in Figure 3-5. The angle is found by taking the angle of the locus that goes through the origin and is tangent to an individual Mohr circle. Since both the EYL according to Jenike and the real EYL are tangent to a Mohr circle wi th major principal stress 1, it can be shown that the relationship between the e ffective angle of internal friction as defined by Jenike J, the real angle of internal friction r, the inherent cohesion ci, and this major principal stress is given by e quation 3-2. This equation is used for the dotted curve in Figure 3-5. 1 1cos sin cos sin r i r r i Jc c 3-2

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42 30 40 50 60 70 80 90 0123456789101 (kPa) (degrees) 1.3 Figure 3-5: Effective angle of intern al friction according to Jenike (J) as a function of the major principal stress (1) for BCR limestone, measured with a Schulze Cell (diamonds) and according to equation 3-2 (dotted line). The smallest Mohr circle, which has a mi nor principle stress of zero and some positive value for the major principle stress, denotes the limit of the curve and will have an effective angle of internal friction (J ) of 90 The major principal stress value of this limiting Mohr circle is 1.3 kPa for the m easured BCR limestone. From equation 3-2 and Figure 3-5, it can be seen that for large valu es of the major principa l stress both effective angles of internal friction converge. As st ated, that is why ci vil engineers are not concerned about the difference between the two angles. In powder technology, with operating pressures between 0 and 50 kPa, the real effective yield locus (with a possible intercept) should be used as opposed to other mentioned loci.

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43 Flow Function As mentioned in chapter 2, the flow f unction of a powder is very important for process equipment design. It is considered an inherent powder prope rty, which is known to be dependent on temperature, moisture, part icle size distribution a nd other factors. The flow function defines the unc onfined yield strength ( fc) of a powder as a function of the history of the powder. This hi story is described by the ma jor principal stress during preshear. This section shows the influence of moisture on the strength as well as the directional dependence of the strength. This di rectional dependence in direct shear cells, or anisotropy, is partly expl ained as a geometric artifact of the testers and is called geometric anisotropy. Influence of Moisture As discussed in chapter 2 and the last se ction, the cohesion of a powder increases with increasing moisture content. The effective yield data shown in the previous section are from a series of data that is used to measure the streng th of a polystyrene powder as a function of the moisture content. The flow f unctions are shown in Figure 3-6. It can be seen that the flow functions initially increase with increasing moisture content, but after a moisture content of 0.5 % decr ease slightly. This has been shown by Esayanur (2005) and is explained as follows. Initially, the mois ture can form liquid bridges between the particles. When the moisture content increase s up to a point where the water can form a monolayer around the particles, the formation of liquid bridges ceases. The monolayer of water will act as a lubricant, decreasing the friction between the particles, hence lowering the strength.

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44 0 1 2 3 4 5 6 7 0510152025301 (kPa)fc (kPa) 0.6% 0.5% 0.4% 0.3% 0.2% Figure 3-6: Flow function of polystyrene powder for different moisture contents, measured with a Schulze cell. Connecti ng lines are for visual clearance. Directional Dependence As discussed in chapter 2, many research ers have shown that a powder can show directional dependence or anisotropy in certa in properties. For the strength of a powder this means that the flow function has different values dependent on the direction in which it is measured. This has been reported with several testers, but no literature has been found on anisotropic strength measurements w ith the Schulze shear cell. To measure anisotropy in a Schulze cell certified BCR limest one was used. The results for a standard strength test were compared with Akers (1992), which is th e international standard, and Saraber et al. (1991). As can be seen in Figure 3-7, the resu lts are alike considering the accuracy of the testers.

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45 0 1 2 3 4 5 024681012141 (kPa)fc (kPa) Akers Saraber Verwijs Figure 3-7: Comparison of st rength measurements of BCR limestone with the Schulze cell (PERC) with Akers (1992) and Saraber et al. (1991). Results from the standard procedure were compared with an an isotropic procedure. With the anisotropic procedure the direction of failure was reversed from clockwise to counterclockwise. To do this, tension rods were designed that can be used under compression. Since the load cells of the Schul ze cells are to be used both in compression and tension a calibration graph was produced for both directions (see appendix A). The results of the experiments are shown in Figure 3-8. It can be seen that the strength of the powder in the direction opposite to the pre-shear is le ss than that in the standard forward direction. When lines ar e fitted through the data and those lines are extrapolated they seem to converge at the yaxis. This is to be expected, since there should be no difference when th e pre-shear stress is zero, i. e., when there is no pre-shear. The pre-shear incorporates a history in the powder and without this history there is no

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46 difference between the two tests. This was also observed by Janssen et al. (2005) for similar tests with a biaxial tester. When a least squares fit is performed with a common intercept at the y-axis, this intercept is 0.71 kPa and the slopes of the linear fits change from 0.26 and 0.11 to 0.28 and 0.09 respectively. 0 1 2 3 4 02468101 (kPa)fc (kPa) Standard Reversefc, standard = 0.26 1 + 0.79 fc, reverse = 0.11 1 + 0.60 Figure 3-8: Flow function of BCR Limestone for a standa rd and reverse experiment, measured with a Schulze cell, with 95% confidence bands for the linear fits and extrapolation to 1 = 0. Angle of Internal Friction The angle of internal friction usually varies with the applied stress. This means that the yield loci (shown in Figure 2-2) are not exactly perp endicular, as shown in Figure 3-9. The behavior of the internal friction a ngle as a function of th e major principal stress appears to be different depending on the dir ection of shear during failure. In the case where the pre-shear and failure are in the sa me direction the angle of internal friction decreases from 38.5 at zero consolidation stress to an angl e around 30 at higher

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47 consolidation stresses. Conversely, in the cas e where the pre-shear and failure were in opposite directions the angle of internal friction increases from 38.3 at zero consolidation stress to about 45 at highe r consolidation stresses. The difference in behavior is explained by the di fference in local stress patterns in the cell during failure. 20 25 30 35 40 45 50 0123456789101 (kPa)() Standard Reverse,standard = -0.88*1 + 38.5,reverse = 0.65*1 + 38.3 Figure 3-9: The angle of internal friction () as a function of the ma jor principal stress for a standard and reverse experiment, w ith 95% confidence bands for the linear fits and extrapolation to 1 = 0. Just like the unconfined yield strength, the angle of internal fric tion for the standard and reverse experiments seems to converge at 1 is zero, i.e., when there is no history in the powder. The value at which they converge is the same value as th e effective angle of internal friction. This shows that for very low stresses the yield locus coincides with the EYL. Therefore, for these values it is very important that the real EYL is used so the correct value for the effective angl e of internal friction is found.

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48 Normal Stress Distribution The Tekscan technique is used to measur e the normal stress distribution at the bottom of the Schulze tester. For this, a Teks can sensor is glued to the bottom of a Schulze cell. A layer of sand is glued to the su rface of the sensor to prevent slip of the powder. Due to the sensor geometry only 21% of the cell bottom can be covered with the sensor. Figure 3-10 shows the normal force distribut ion of a section of the Schulze cell. It can be seen that the force in the center of the angular cell is larger than at the sides near the walls, which is to be exp ected (Bilgili et al., 2004). Figure 3-10: Typical normal stress profile at the bottom of a sli ce of the Schulze cell. The force is not constant in the angular direction of the cell. To show this, the average force in the rectangular window in Figure 3-10 is re corded during a full rotation. The results for three identical tests are plotted versus the angle of rotation in Figure 3-11. It can be seen that the three experiments s how the same trend. There are three to four areas of high force. This would suggest th at these areas mainly support the top lid and load, since three or four point s give a stable support.

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49 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 090180270360 Cell Rotation (degree)Force (N) Figure 3-11: Normal force on the botto m of a Schulze cell during steady state deformation of BCR limestone. Geometric Anisotropy It is believed that this fo rce distribution is the cause of a geometric anisotropy, i.e., due to the tester geometry. This is explai ned by the following fi ndings. Bilgili et al. (2004) presented measurements of local stress es in the Jenike cell using Tekscan pads. They found a skewed pressure profile in the Jenike cell w ith a higher stress toward the leading edge of the cell. The same techni que was used to measure anisotropy in the Jenike cell. A failure in the same direction as pre-shear was compar ed with a failure in the direction 180 degrees rotated from the preshear. This procedure is similar to that used by Saraber et al. (19 91) in their anisotropic Je nike shear measurements. Figure 3-12 shows the pressure profiles m easured at the cell bottom during the case of pre-shear in the forward di rection and failure in the forw ard direction. The pre-shear

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50 profile shows the typical skew ed profile observed by Bilgili et al. The failure profile shows the same skewed behavior but at lowe r normal pressures. Figure 3-13 shows the normal pressure measured at the bottom of the test cell for the conditi ons of pre-shear in the forward direction and failure in the reverse direction. Th e same skewed profile exists in the pre-shear profile and the failure profile, except that the failure profile is skewed in the opposite direction. Because of this the lo cal normal stress values during failure are higher than the local normal stress during th e pre-shear for approximately 1/3 of the cross-sectional area. This s uggests that a portion of the ma terial in the test cell is undergoing a transition of stress state during the failure mode of this test. The failure state during reverse shear is a combination of act ual failure conditions in a portion of the material and attainment of a new lower steady st ate shear stress in the rest of the material. This will obviously result in a lower overall shear stress value, which is measured for anisotropic experiments. Sara ber et al. suggested a renewe d steady state to explain the lower failure values for anisotropic shear. Thes e new data indicate that Saraber et al. were half right. The anisotropic behavior induced in the Jenike cell is due to the attainment of a new critical state of stress, but occu rs only in a portion of the cell. A similar geometric anisotr opy is expected in the Schulze shear cell. The results show that there is an angular stress distributi on which is similar to a skewed profile in the Jenike cell. Since not the entire bottom of the Schulze cell is covered no comparison between the steady state and the failure state can be made. Failure occurs instantaneously and with only 21% of the cell bottom covere d with the Tekscan pad, no conclusive measurements can be made.

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51 Steady State shear direction Failure fail direction 0 0.5 1 1.5 2 2.5 3 3.5 0 20 40 60 80 100 120 Distance Across Cell (mm) Normal Stress (kPa) Stead y State A t Failure Figure 3-13: Stress profile at the bottom of a Jenike cell dur ing pre-shear and failure in the forward direction, using silica powder. Steady State shear direction Failure fail direction 0 0.5 1 1.5 2 2.5 3 0 20 40 60 80 100 120 Distance Across Cell (mm) Normal Stress (kPa) Steady State A t Failure region where fail stress exceeds pre-shear stress Figure 3-13: Stress profile at the bottom of a Jenike cell dur ing pre-shear in the forward direction and failure in the backwa rd direction, using silica powder.

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52 Modeling of Anisotropy in Biaxial Shear This section describes work that has been done in collaboration with Dr. Zhupanska. The anisotropy that is measured in a biaxial shea r tester is compared with DEM simulations. The experiments where pe rformed with the Wall Flexible Biaxial Tester (FWBT) (van der Kraan, 1996). This test er is an advanced pow der flow tester that allows for controlling stress and strain i ndependently in two mutually perpendicular directions. As a result, stress-strain curves for powder as well as unconfined yield stress can be measured. The experimental study en compasses the comparison of standard and anisotropic flow functions, similar to the test s as described above. It should be noted though that no direct comparison is possible since the Schulze tester is a direct shear tester and the FWBT is an indirect tester. Biaxial Experiments The experimental procedure of the tests as well as an extensive description of the tester can be found in Janssen (2001) and Verw ijs (2001). In short, the unconfined yield stress is measured in the following way consis ting of four steps. During the first step, the powder is biaxially consolidated up to a chos en stress value. From this point, the powder is pre-sheared (step two) by in creasing the stress in one dire ction and decreasing it in the perpendicular direction. During this step the mean stress is kept constant. At a certain moment the powder is in steady state defo rmation and the stress cannot be increased anymore. When this steady state deformation is reached, the third step is a relaxation of the stresses to near zero. The fi nal step is an increase of stress in the direction of major stress during pre-shear while the stress in th e other direction is kept zero. The maximum stress that causes sample failure is the unconf ined yield stress. A typical test result for the standard experiment is shown in Figure 3-13.

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53 In anisotropic experiments, the first thr ee steps are the same but during the final step the stress is increased in the directi on of minor stress during pre-shear while the stress in the direction that wa s major during pre-shear is kept at zero. So the stress profile was rotated 90 degrees with respect to the sta ndard experiment during step four. A typical result of the anisotropic experiments is s hown in Figure 3-15. It can be seen that anisotropic failure takes more time and will result in a lower value. 0 2 4 6 8 10 05101520time [h][kPa]flow1 2 3 4 failure Figure 3-14: Typical result of a standard experiment with the Flexible Wall Biaxial Tester, showing the stress in the x-di rection (black) and y-direction (grey), and the different steps of the test; biaxial consolidation (1), pre-shear (2), relaxation (3), and failure (4). 0 4 8 12 16 05101520time [h][kPa]anisotropic flow1 3 4 anisotropic failure 2 Figure 3-15: Typical result of an anisotropic experiment with the Flexible Wall Biaxial Tester, showing the stress in the x-di rection (black) and y-direction (grey), and the different steps of the test; biaxial consolidation (1), pre-shear (2), relaxation (3), and anis otropic failure (4).

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54 Set-up Biaxial Simulation The biaxial tests were simulated in 2D, om itting the z-direction, using the Itasca 2D DEM code. Following the experimental procedure, a sample of size 0.075 0.075 m was chosen. The simulations were done for about 7427 uniform circular discs. Disc size was 0.45 mm. A linear stiffness model was used for particle-particle and particle-wall interactions with normal and tange ntial stiffness equal to 1.0*106 N/m for the particles and 1.0*108 N/m for the walls. The friction coefficient was 0.577 for particles and 1.0*10-5 for the walls. The low friction coefficient for the walls was chosen since the FWBT is designed to have no wall friction. Particle density was set to 2650 kg/m3 and the initial poros ity was set at 0.16. Strain control was set a wall velocity of 0.00001 m/s. Following the real experiments in the biaxia l tester described above, four different stages were simulated with the DEM code (see Figure 3-16). The first step, biaxial consolidation, was performed by uniform moti on of all four walls towards the center of the sample. The second step, pre-shear, wa s simulated by continuous motion of the xwalls towards each other and motion of the y-walls away from each other. Just as in the real experiment, at a certain moment, the samp le started to deform with a steady state, and stresses on the walls did not change anymore. At this point the third step, relaxation, was invoked on the powder by releasing the walls until the stresses on the walls were almost zero. As with the FWBT experiment s, the fourth step was performed in two different ways. For the standard experiment the x-walls were moved in the direction of pre-shear until failure, while the y-walls did not move. For the anisotropic experiments the y-walls were moved towards the center of the sample while the x-walls remained stationary.

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55 biaxial consolidation pre-shear relaxation normal failure anisotropic failure Figure 3-16: The direction of wall movement at different stages of the biaxial test simulation. Stress and Porosity Distributions During the different deformation steps th e changes in stress and microstructure were monitored. Figure 3-17 shows the norma l stress in the x and y direction and the porosity at the end of a biaxia l consolidation to 17.2 kPa wall stress. It can be seen that the stresses develop patterns sy mmetric about the xand y-ax is. The stress distributions are not uniform as is generally assumed for biaxial consolidation. The magnitudes of the stresses in the x and y direction are comp arable, between 15 and 18 kPa. The porosity distribution in the sample is symmetric with respect to the symmetrically applied load, but there is a significa nt difference between the porosity in the center and at the walls. 0.2 0.1 18kPa 15kPa 18kPa 15kPa xx yy porosity Figure 3-17: Normal stresses and porosity dist ributions after biaxial consolidation to 17.2 kPa.

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56 The normal stress distributions and poros ity after the second deformation step, the pre-shear, are shown in Figure 318. It is apparent that the stresses in the x-direction are twice as large as the stresses in the y-direction. This is to be expected since the sample is loaded in the x-direction. There is a signifi cant change in structure noticeable compared with the biaxial consolidation. The porosity distribution is highly non-uniform in the direction of compression but rela tively uniform in the direction of relaxation. From this it can be concluded that pre-shear causes significan t structural anisotropy that will lead to anisotropic behavior in subsequent steps. 0.22 0.12 12kPa8kPa 26kPa 19kPa x x yy porosity Figure 3-18: Normal stresses and poros ity distributions after pre-shear. After steady-state deformation, the walls were retracted until the stresses on the walls were close to zero. A zero stress level on the walls is reached when practically all contact forces are equal to zero at the particle contacts. Because of this, the stress distributions as shown in Figure 3-18 disi ntegrated and became very homogeneous, i.e., all zero. The porosity increases slightly dur ing relaxation but the pattern remains the same as in Figure 3-18. Since the porosity di stribution represents the sample structure, relaxation of the sample cannot remove stru ctural anisotropy introduced during the preshear. Figures 3-19 and 3-20 show the results of the fourth step, failure in the same direction as pre-shear and perpendicular to that, respectively. As expected, a different response was recorded during this step, depe nding on the direction of shear. When the

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57 failure was performed in the direction of pre-shear the porosity distribution remained practically the same, while the overall porosity sl ightly increased. In the case of failure perpendicular to the direc tion during pre-shear a signi ficant change in porosity distribution can be seen. The overall porosity decreases slig htly, while the distribution becomes symmetrical, comparable to the distribution after consolidation. The stresses in Figure 3-19, the standard experiment, are clear ly higher than the st resses in Figure 3-20, the anisotropic experiment. This is in agreem ent with the experiments performed with the FWBT. It appears that there is a diagonal shear zone during failu re of the standard simulation, as can be seen most clearly from the xx plot in Figure 3-19. The anisotropic simulation does not show such a clear zone. This seems to be in agreement with the hypothesis proposed in the secti on above. During a standard e xperiment there is a real failure, but during an anisotropic failure there is a change to a stress st ate that represents a (partial) new steady state deformation. This is also confirmed with the porosity distributions. The standard simulation does not show a change of structure, while the anisotropic simulation shows a cl ear change of structure. It is expected that if the anisotropic simulation is taken further in time, the anisotropy distribution becomes similar to that during pre-shear with 90 degr ees rotation. The porosity in Figure 3-19 does not show a higher porosity for the diagonal shea r zone, as would be expected. This could be due to a limited resolution of the porosity calculation. It should be noted that the biaxial experiments were stress controlled wh ile the simulations were strain controlled. The effect of stress and strain control will be discussed in chapter 3.

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58 0.3 0.1 16kPa10kPa 30kPa18kPaxx yy porosity Figure 3-19: Normal stresses and porosity distributions after forward failure. 0.20.1 18kPa 27kPa 11.5kPa9.5kPa xx yy porosity Figure 3-20: Normal stresses and porosity distributions after anisotropic failure. Microstructure The distributions as shown above can also be studied at the partic le or micro scale. To describe the structure of the particles the concept of fabric measures generally accepted in the field of soil mechanics can be adopted. This enables the investigation of the effects of the microscopic structure on the macroscopic powder behavior, e.g., on the development of anisotropy. To do this two fa bric measures were calculated. One fabric measure was related to the dist ribution of the contact normal s and one was related to the intensity distributions of the contact forces In the 2D case these distributions can be represented by an angular di stribution that defines th e fraction of the measure K( ) in a certain direction as shown in Figure 2-13. In practic e the distribution is discretized in sections If the distribution function is circular there is no preferenti al direction of the measure and the (overall) sample is isotropic for that measure.

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59 The intensity of the distri bution is determined by numb er in the case of contact normals and by the total magnitude in the case of contact forces. In this study the overall distribution for the sample has been studied. If the number of particles in a simulation is large enough it is also possible to evaluate fa bric measures for different positions in the sample. The changes in the fabric measures were m onitored at the end of every step of the simulation of the biaxial test. Slight changes were noticed in the or ientation of contact normals from a fully isotropic distribution after biaxial consolidation to a slightly asymmetric distribution after relaxation. It is concluded that in the simulations no significant changes in the orient ation of contact normals coul d be observed. This is in contrast to other studies (Oda et al., 1980; Rothenburg and Ba thurst, 1989) where particle contacts were preferentially aligned in the dir ection of the applied major principal stress. consolidation p re-shear normal failure a nisotro p ic failure Figure 3-21: Normal contact force intensit y distribution after different steps of the biaxial simulation. The normal force distributions showed a clear difference between the different simulation steps (see Figure 3-21). The dist ribution of the normal contact forces was asymmetric and oriented in the direction of th e major compressive stress after every step

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60 except for the biaxial consolidation. There is no normal contact for ce distribution for the relaxation step since all normal contact fo rces were near zero. Since friction was practically absent on the walls of the cell, the tangential contact forces were two orders of magnitude smaller than the normal contact fo rces and did not cont ribute significantly to the anisotropic response. It should be noted that a simplified m odel is used for the simulations, i.e., the simulation is 2D, the partic le contact model is a simple linear elastic model, the particles are all spherical and of equal size. Therefore, the results can not be used quantitatively. It is cl ear though that qualita tively the simulati ons show anisotropy that is similar to the anis otropy that is observed in the biaxial experiments.

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61 CHAPTER 4 UNIAXIAL COMPACTION OF POWDERS Uniaxial compaction or compression is a proc ess that is largely u sed in industry to make compacts like tablets or briquettes. There are roughly two operation regions when considering compaction. There is a high stress region, used for tabletting, and a low stress region, used for powder flow evaluation. Th e stresses used in the high stress region are several magnitudes higher than the stresses encountered in the low stress region. This chapter concerns with the low stress region, a lthough some of the fi ndings are applicable in high stress compaction. The main difference is that the stresses enc ountered in the flow regime are generally not high enough to pe rmanently deform the particles, unless the particles are very fragile. Also the porosity in the flow regime is higher than in the high stress compaction regime. There are several uniaxial testers availa ble commercially for the measurement of compaction curves. For this research a new uniaxial tester has been developed. The unique feature of the tester is its capability to deform a pow der either stress or strain controlled. It is generally accepted that str ess and strain are dire ctly and reversibly related, making it irrelevant whet her a tester is stress or stra in controlled. This research shows that this is not true for all powders The compaction stress operating window of the tester is zero to 35 kPa. The tester is built as a prototype for the development of a true triaxial powder tester with independent stress an d strain control in a ll principal directions.

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62 Design of the Uniaxial Tester The uniaxial tester consists of two main parts, the actuator system and the box that contains the powder sample. The actuator syst em consists of two identical actuator units, working in line and in opposite direction. The actuator system changes the sample holder size to deform the powder. Figure 4-1 s hows a picture of th e uniaxial tester. A F E D C B G A F E D C B G Figure 4-1: Picture of the Unia xial Tester showing servomotor (A), linear stage (B), aircylinder (C), sample holder box (D), linear air bearing (E), guide bar (F), and pressure regulator (G). Each actuator unit consists of a linear stage for strain controlled motion and an aircylinder for stress controlled motion. Th e linear stage is a 100 mm wide 118 series American Linear Motion (model PGA4-9-4/GB) position stage. It h as 100 mm travel via a 10 mm OD x 2 mm precision lead screw. The accuracy of the stage is 25 m/m. The stage is driven by a Cool Muscle servomot or (model CM1-C-23L20) via a flexible coupling. The motor has a high-resolution encoder of 50,000 units per rotation. The motor has smooth motion even at low speeds. The air-cylinders are custom made. They ar e drilled out of a r ectangular aluminum block and have a high accuracy 44.45 mm bor e diameter. The pistons are made of white Delrin. To get near frictionless motion the pist ons are not sealed. The pistons are tapered and the smallest clearance be tween the piston and the cylinde r is about 0.05 mm, making

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63 the cylinders leak. The cylinders are theref ore called “leaking cylinders”. The leaking airflow lubricates the piston, maki ng the piston near frictionless. The leaking cylinders are pressurized with two independent pressure regulators (Marsh Bellofram, type 3211). The range of the regulators is 0 – 350 kPa gauge, corresponding to an actuator force of 0 – 543 N. The max flow of the regulators is 12 SCFM, which is enough to pressurize the leaki ng cylinders up to the maximum pressure. The air-cylinders are mounted on the lin ear stages. The sample holder box is connected with the two actuator units via sh afts. A shaft includes a load cell, measuring the applied load. The shaft can be locked fo r strain controlled e xperiments and unlocked for stress controlled experiments. To st abilize and support the sample holder box there are two guide bars per actuat or unit. All guide bars are supported by two linear air bearings. These air bearings provide a fric tionless linear motion and are stiff enough to support the guide bars and sample holder box. Fi gure 4-2 shows a top vi ew picture of the system showing the guide bars and the shafts. E D C B A E D C B A Figure 4-2: Top view picture of the Uniaxial Tester (without top lid) showing the aircylinder (A), shaft (B), linear air bearing (C), guide bar (D), and sample holder box (E).

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64 The sample holder box is an important part of the tester. Due to friction at the walls, the geometry of the sample holder infl uences the test results The design of the uniaxial tester with a separate actuator sy stem and sample holder allows the use of different box designs. The diffe rent box designs that are built are rectangular in shape. The box design can also be cylindrical but that would make the filling process complicated. Figure 4-3 shows the different rectangular box designs. Two Half Boxes Two U-Boxes Channel Two Half Boxes Two U-Boxes Channel Figure 4-3: Diffe rent sample holder box designs. The difference between the boxes is the dire ction of the shear stress at the side, bottom, and top walls. In the “two half box es” design (also shown in Figure 4-2 without the top), the shear stress at the bottom and one of the side walls is in the same direction, while at the top and the othe r side wall it is in the opposit e direction. In the “two Uboxes” design the shear stress is in the same directions at th e top and bottom wall, and in the opposite direction at the si dewalls. In the “channel” design the shear stresses are in the same direction at all walls when only one piston is m oving. The stress directions change approximately midway of the bottom, top and side walls when both pistons are moving.

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65 The position of the box parts is monitore d with two linear potentiometers (Omega model LP804). The potentiometers are mounted on the outside of the air-cylinders and attached to the back of the box. This enab les monitoring of the position of the cylinder piston and calculation of the sample size. Th e two loading walls both have a constant cross sectional area of 127 x 127 mm2. This gives a ratio of th e pressure in the leaking cylinder over the stress on the powder of 10.4. Uniaxial Experiments Initial tests with the uniaxial tester s howed that the two half-boxes sample holder design could not be used at high stresses. Wh en a powder is uniaxially compacted, there is a resulting normal stress at the side wa lls. In the two half-b oxes design, the normal stress on the side walls causes a moment on the air bearing supporting the sample holder. At a compacting stress above about 8 kPa th e moment on the bearings becomes larger than their rating. This causes friction in th e bearings, which can damage them. It also causes hysteresis in loading-unloading loops, affecting the measurements. In the two uboxes design, the moment of opposite walls is approximately offset, avoiding the problem. The experiments reported in this disser tation are performed with the two u-boxes sample holder. The tester is filled by scoopi ng powder through a sieve into the sample holder box. The sieve breaks up a ny agglomerates and ensures a more or less consistent packing from sample to sample. The excess po wder is scraped off from the top and the top lid is placed. Two types of experiments are performed; stress controlled and strain controlled. Stress controlled means that the tester dictat es the stress path and the resultant powder deformation is measured. For the experiments re ported the stress path is set at a constant

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66 stress increase on the powder. Strain cont rolled means that the tester dictates the deformation path and the stress response of the powder is measured. For the experiments shown the strain path is set at a constant strain velocity. For both types of experiment, one actuator unit is locked, resulting in uni axial consolidation w ith one moving wall. The two powders that are tested are Mi crocrystalline Cellul ose powder (PH101) and Polystyrene powder. The mean particle size of Microcrystalline Cellulose is 50 m. The mean particle size of the Polystyrene powder is 40 m with a narrow size distribution (see appendix B). Uniaxial Compaction of Microcrystalline Cellulose Microcrystalline cellulose is compacted stre ss controlled at a rate of 0.01 kPa/s and strain controlled at a rate of 0.48 mm/min. The resulting co mpression curves are shown in Figure 4-4. It can be seen th at the two curves do not differ si gnificantly. That is what is mostly reported in literature. That is why it is generally assumed that stress control and strain control produce states th at are indistinguishable. For operation of process equipment it do es matter though whether stress control or strain control is used to run a process. This is due to the sh ape of the compaction curve. It can be seen that the stress increases exponentially. So for high stresses the stress can double with a minimum deformation. In a ta blet press the stamp and die are strain controlled and set at a certai n deformation. A minimal de viation in powder load will therefore result in a significant compacti on stress difference. The compaction stress defines the strength of the tablet. If for quality purposes the tablets need to have a controlled strength, e.g., pharmaceutical pills, th e press should be stress controlled. If the

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67 size of the tablets is the foremost concern, e.g., precision cast parts, the press should be strain controlled. 0 4 8 12 16 20 00.040.080.120.16 Strain (-)Stress (kPa) stress control strain control Figure 4-4: Uniaxial compacti on of Microcrystalline Cellul ose PH101, stress controlled (grey) and strain controlled (black). Uniaxial Compaction of Polystyrene Powder Polystyrene is a powder that exhibits stick-slip. The powder is stress controlled at a rate of 0.045 kPa/s and strain controlled at a rate of 0.2 mm/s and the results are shown in Figure 4-5 and 4-6, respectivel y. The difference between Fi gure 4-4 and Figures 4-5 and 4-6 is apparent. The stress-stra in curve for microcrystalline cellulose is smooth while the stress-strain curves for polystyrene ar e very jagged, showing stick-slip. There is a clear difference between the stre ss controlled and stra in controlled stressstrain curves of polystyrene. In the str ess controlled experiments the slips exhibit themselves by quick deformations, plotted as near horizontal lines in the curve.

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68 0 5 10 15 20 25 00.020.040.060.080.10.120.14 Strain (-)Stress (kPa) Figure 4-5: Stress controlled uniaxial compaction and relaxation curve of 40 m polystyrene powder. 0 5 10 15 20 25 3000.020.040.060.080.1 Strain (-)Stress (kPa) Figure 4-6: Strain c ontrolled uniaxial compaction curve of 40 m polystyrene powder.

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69 The sticks on the other hand endure a long tim e and form near vertical lines in the curve. In the stress controlled experiment ther e are few, but very violent stick-slip events. Due to the fast slip in which the powder de forms up to a few millimeters in a fraction of a second, powder is blown out of the sample hol der. The air that is present in the powder voids has to escape when the sample deforms. Th e tester walls are solid, so the air has to escape through the narrow gaps between the walls. When this happens very fast, the velocity of the air through the gaps becom es high and the air will entrain some of the powder. In the strain controlled experiment a slip exhibits itself by a fast stress decrease, shown as a vertical line in the curve. Duri ng the stick phase the stress increases until the next slip. The stick-slip events are very fr equent but due to the controlled deformation rate the events are not as violent as the stress c ontrolled slip events. The stick-slip signal is investigated in detail in chapter 5. The experiments with polystyrene show that stress control and strain control are not reversible for certain powders. Starch powde r showed the same behavior as polystyrene. The conclusion of the work is that powders th at show stick-slip wi ll behave differently when stress controlled or strain controlled, making the two control systems non-identical. Modeling of Uniaxial Compaction Using the Itasca PFC 2D Discrete Elemen t Method the uniaxial compaction is investigated. The focus was on the influen ce of the sample holder geometry and the influence of stress and strain control on the powder structure. For the simulations a linear contact stiffness model for circ ular particles is used. The 10,400 simulated particles range in size from 80 m to 1.2 mm and have a uniform dist ribution. The friction coefficient is

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70 taken to be 0.577 between the particles and 0.364 between the particles and walls. These values are estimates but it will be shown in chapter 6 that the value of the friction coefficient should be measured accurately sin ce it is of major impor tance. The cell size for the simulations is 63.5 mm x 63.5 mm and the initial 2D porosity is 0.17. Influence of Cell Geometry As discussed at the beginning of this chapte r, the direction of the shear stress on the walls will be different for di fferent sample holder geometries. The stress distribution and structure of the particle ensemble in the tw o half boxes and the tw o u-boxes design of the uniaxial tester are investigat ed. The two half-boxes design is called geometry 1 and the two u-boxes design is called geometry 2 and th ey are shown in Figure 4-6. It should be noted that geometry 2 is a cross-section of th e center of the u-boxes design and represents more correctly the 2D simulation of a channel design with one moving wall. Figure 4.6: Sample box geometries used for th e simulations; geometry 1 (left) simulates the two half-boxes design, geometry 2 (right) simulates the two u-boxes or channel design. The initial particle ensemble is th e same for both simulations. During the simulation of geometry 1 the left and bottom wall move to the right while the other two walls are stagnant. During the simulation of geometry 2 only the left wall moves to the right while the three other walls are stagnant. The geometries are both strain controlled deformed with a strain rate of 9*10-11 step-1. The time scale is given in steps and not in

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71 real time since it reduces the computation time. In Figure 4-7 the stresses on the xand ywalls is given for the two geometries as a func tion of the time steps. In this case the xaxis is equivalent to the stra in. It can be seen that there is no significant difference in the overall wall stresses for the two geometries. 0 100 200 300 400 500 600 0.00E+008.00E+061.60E+072.40E+07 Time (Step)Stress (Pa) Geometry 1, x-walls Geometry 1, y-walls Geometry 2, x-walls Geometry 2, y-walls Figure 4-7: Overall st ress on the xand y-walls of two cell geometri es during uniaxial compaction. The stress distributions in the particle ensembles do show a difference between the two geometries. The difference of the normal stress distribution in the y-direction is minimal but the normal stress distribution in th e x-direction is signifi cant. This is shown in Figure 4-8. Geometry 1 shows high stress region diagonally from the top left to the bottom right while geometry 2 shows a more symmetric stress dist ribution with a high stress band in the middle from left to right with high stress points in the left corners. This

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72 could indicate that there is a shear plane in geometry 1, something that is generally considered not possible in a uniaxial tester. Figure 4-8: Distribu tions of normal stress in the x-di rection for geometry 1 (left) and geometry 2 (right). The unit of the scal e is kPa but for a 2D simulation. In Figure 4-9 the fabric tensors are show n for the two simulations. It can be seen that the angular distribution of the contact normals is isotropi c. This means that there is no preferential contact direction, similar to the results in the previous chapter. There is a clear anisotropic distribution of the magnit ude of the normal forces. The difference in anisotropy between the two different geomet ries is not significant though. The magnitude is slightly larger for geometry two, but th is is because this si mulation is stopped at a slightly higher stress level. The difference be tween the two simulations is apparent in the angular distribution of the ta ngential contact forc es. Surprisingly, geometry 1 is more isotropic than geometry 2. It should be noted that the magnit ude of the tangential forces is fifty times smaller than the magnitude of the normal forces. Therefore, the influence on the overall bulk behavior will be less significan t. This would exclude the possibility of shear zones during powder compaction.

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73 -750 750 -750750 Number of Contacts -750 750 -750750 Number of Contacts -200 200 -200200 Magnitude Normal Contact Force (N) -230 230 -230230 Magnitude Normal Contact Force (N) -4 4 -44 Magnitude Tangential Contact Force (N) -4 4 -44 Magnitude Tangential Contact Force (N) Figure 4-9: Fabric te nsors for uniaxial compaction usi ng two different geometries. The tensors at the left are for geometry 1 and the tensors at the right are for geometry 2. Shown are the angular di stributions of the particle contact normals (top), normal force magnit ude (center), and tangential force magnitude (bottom).

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74 It should be noted that th e fabric tensors in this work are used as overall parameters, covering the whole sample. This means that there could be a narrow shear zone with a high shear stress th at is not captured because the volume of the shear zone (or area in 2D) is much smaller than the overall volume. Ther efore to reach conclusive evidence of shear zones during powder compact ion it is necessary to use local fabric tensors to identify the structure in different regions in the powder to investigate possible shear regions. For that a larger number of pa rticles is necessary to make a statistically correct angular distribution. Pseudo-Stress Control versus Strain Control The experiments with the uniax ial tester showed that fo r certain powders stress and strain control give distinctively different r esults. Some testers a nd process equipment are said to be stress controlled while they actua lly are pseudo stress controlled. Pseudo stress control is indirect control. The stress (or us ually load) is measured with a sensor. Through a feedback control the stress on the powder is maintained by a strain device, usually a motor with a lead screw. If the stress is too low, the moto r moves the walls inwards, and if the stress is too high, the motor moves the walls outwards. This is called servo control. Measurements with pseudo stress control can be seriously biased. The feedback control needs to be much faster than the processes in the powder. Since we do not know the fundamental powder behavior, we do not kno w for sure that our feedback is fast enough. It is very well possible that test er artifacts are measure d instead of powder properties. It is not very like ly that a pseudo stress controlle d tester can measure stick-slip correctly. The slips in th e powder are too rapid. To investigate the influence of pseudo st ress control, uniaxial simulations are performed using the servo control method. Si nce the Itasca code do es not provide a real

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75 stress controlled mechanism the servo results are compared with pure strain control. The simulations are performed using geometry 1. The two gain f actors that are investigated are 0.001 and 0.005. A larger gain is faster enabling a quick res ponse, but it is also coarser. When controlling a powder, the large gain can cause significant overshoots. The results are shown in Figure 4-10. As expected, the larger gain results in a larger initial control step. The stress quickly incre ases to about 100 Pa for a gain of 0.005. After some strain the two pseudo stress controlled curves seem to converge. This would mean that only for low stresses pseudo stress cont rol biases the results. This initial difference might not be large enough to ch ange the particle structure to cause different behavior. It is likely that it will largely influence the behavior for a powder showing stick-slip. The difference in stress-strain curves is a due to a combination of test er characteristics and powder characteristics. 0 100 200 300 400 500 600 00.00050.0010.00150.0020.0025 Strain (-)Stress (Pa) pseudo stress control (0.005) pseudo stress control (0.001) strain control Figure 4-10: Consolidation curves for uniaxial simulations; strain controlled (red) and pseudo stress controlled with a gain fact or of 0.005 (blue) and 0.001 (black).

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76 CHAPTER 5 STICK-SLIP IN POWDER FLOW Some powders exhibit stickslip during deformation but most do not. A very well known powder that exhibits stick-slip is snow. When we compress snow to make a snowball or snowman, but even more when we walk on snow, we can feel and hear the stick-slip. The snow does not deform in a smooth manner but in many short steps, quickly following each other. These steps cause the crunching sound that we hear. In this research snow was not chosen as a model pow der. Besides expensive climate control, snow particles or flakes are fractals, which makes the modeling of the system very complex. In the kitchen we can find two powders, flour and starch, that initially seem very similar. Starch is slightly whiter than flour, which enables us to see the difference. When we feel the powders they appear very differe nt though. Starch exhibits stick-slip, which we can feel and hear when we squeeze it be tween our fingers. Flour does not show this phenomenon. This chapter investigates the macroscopic response of stick-slip in powders. Both the magnitude and frequency of stick-slip ar e investigated as a f unction of both operating parameters and powder properties. The two ope rating parameters that are varied are the normal force that is exerted on the powder and the deformation rate or shear velocity. The powder parameters that are varied are the particle size distribution and the moisture content.

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77 Introduction to Stick-Slip In the field of tribology mu ch research has been conducte d on stick-slip between surfaces. Stick-slip is a non-smooth motion, in jerks, and can be regular (periodic) or irregular (erratic or intermittent). It is ge nerally considered that the difference between static friction and kinetic friction causes the material to move intermittently (Bowden and Tabor, 1950). The difference between static a nd kinetic friction is due to the surface roughness. The microscopic contact area betw een surfaces will be larger in static conditions than during slid ing, resulting in a high er static friction. Generally, lubrication reduces or eliminat es stick-slip but even lubricated surfaces can exhibit stick-slip. Thompson and R obbins (1990) did molecular dynamics simulations of a lubrication layer between two plates. The bottom plate was kept stationary while the top plate is moved att aching it with a spring to a translating stage with a constant velocity. They attributed the stick-slip in the boundary lubr ication to a transition from a crystalline state of the lubrican t to a fluid state. A thin layer of lubricant can form a structure that is ordered like a cr ystal, which periodically melts due to the shear between the plates, causi ng the slip, and recrystallizes, causing the stick. The results indicate that stick-s lip in boundary layers is caused by thermodynamic instabilities during the sliding phase rather than a dynamic inst ability. It is observed that stick-slip is a function of the shear velocity. The frequency of stick-slip increases generally with the velocity. Berman et al. (1996) present three differen t stick-slip mechanisms. The first is the surface topology or roughness model as depicted in Figure 5-1. Due to the asperities on the contacting surfaces, the friction force in creases when an asper ity on one surface has to travel over an asperity on the opposite su rface. When the two asp erities have passed

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78 each other, the friction force will quickly drop due to the quick fall, e.g., from B to C. Stick-slip due to surface r oughness is irregular, unless th e surface roughness has a regular pattern. The stick-slip frequency increases with the shear velocity while the amplitude decreases. A Time Friction Force A B B C C D D E E Figure 5-1: Schematic representation of stick-slip due to surface roughness. The second stick-slip mechanism is calle d distance-dependent model or creep model. During shearing, due to the adhesion between asperities, th e surfaces have to creep a distinct distance before the adhesi on breaks and the surfaces slip. This type of stick-slip is mainly observed for dry systems a nd results in a regular, periodic, signal. It is often used to model rock-on-rock slidi ng in geology. The amplitude decreases with increasing shear velocity. The third stick-slip mechanism is the velo city-dependent model. This model applies to lubricated systems. At low velocities stic k-slip is present but it can vanish above a certain critical velocity. From their simulations, Robbi ns and Thompson (1991) found that for lubricated surfaces this critical velocity is equal to the velocity of the top plate just before recrystallization of the lubric ation boundary. In their system the critical velocity is a function of the mass of the top plate and not an intrinsic lubricant property. Yoshizawa and Israelachvili (1993) perfor med similar simulations for different

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79 lubrication species. They found a critical veloc ity dependent on the spring constant of the spring pulling the top plate. So the critical stick-slip velocity be tween two surfaces is a combination of the surface and lubricati on properties as well as the test system. The conventional models generally cannot e xplain stick-slip in granular material. Many phenomena are similar for both systems though. Friction in gran ular material can be regular and irregular. A critical velocity is found fo r both friction systems. The dependency of the friction for ce on the velocity is often re presented as a bifurcation as depicted in Figure 5-2 (Buck lin et al., 1996; Kolb et al ., 1999; Lacombe et al., 2000) Friction Force F max Time Shear Velocit y Friction Force F min vi vc Figure 5-2: Schematic representation of the bi furcation of the friction force as a function of the shear velocity. The bifurcation point is the critical stick-slip velocity vc. Budny (1979) published what might be the fi rst publication about stick-slip in powders. He used stick-slip as an alternative method to characterize powder flow. His experimental set-up comprised of an Instrom pulling an aluminum plate over a table. The table and aluminum plate were covered with sandpaper to provi de friction. In between the plates a thin bed of powder was placed. The model Budny proposed to describe stick-slip is a modified Mohr-Coulomb model, shown in equation 5-1. The model can be used to find the internal frictional coefficient i as well as a stick-slip friction coefficient s as a function of the normal force Fn and the stick-slip frequency The term sFn is called the stick-slip friction and is the difference between the maximum and minimum friction.

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80 t F F Fn s n i friction sin 5-1 Budny found a decrease in the stick-slip friction, or ma gnitude of the stick-slip, with increasing sled velocity. This is consis tent with Figure 5-2 but he did not report a critical velocity for a range up to 100 inch es per second. He reporte d a decrease in stickslip magnitude with in creasing particle size. Bucklin et al. (1996) used a similar setup to measure stick-slip in a granular wheat bed. They pulled a plate of wall material thr ough a bed of powder, thus both sides of the plate were in contact with the grains while the grain bed was loaded with weights. Bucklin et al. reported a critical stick-slip velocity and a bifurcation similar to Figure 2-5. The lower bound (minimum friction) seemed to be a continuation of the curve without stick-slip, while the upper bound (maximum friction) s eemed to divert from the bifurcation point upward. No dependency on the pressure was observed for the critical velocity. Albert et al. (2001) measured the force on a rod in a rotating granular bed and reported stick-slip. The rod was connected to a load cell with a spring. The spring constant could be changed by exchanging sp rings. Below a certai n spring constant no dependency on the spring constant or the sh ear velocity could be found for the friction force. The product of the spring constant, the velocity and the inve rse of the frequency was constant. Depending on the depth of the rod in the gr anular bed and the particle size, the stick-slip signal was either periodic, random, or stepped (s ee Figure 5-3). Al bert et al. concluded that the stick-slip be havior is due to jamming of the rod in the granular bed. The development of force chains in the bed is dependent on the particle size as well as

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81 the depth of the rod. A linear increase of the magnitude of stick-slip with rod diameter was reported. Figure 5-3: Different types of stick-slip accord ing to Albert et al. (2001). Schulze (2003) used the a ring shear tester to investigate the time and velocity dependency of stick-slip. He s howed that the shear stress of powders can either increase or decrease with increasing sh ear velocity, depending on the powder. When a powder is sheared, stopped for a while, and sheared agai n, either an increase or decrease in shear strength can be measured, again depending on the powder. Schulze accredits this to the creep behavior at the particle contacts. A powder that shows an increase in shear stress after a stop period will show an increase in stick-slip ma gnitude with decreasing shear velocity. A decreasing shear velocity means that the stick time is longer and thus the creep can work longer, creat ing a higher shear stress. Volfson et al. (2004) used molecular dynami cs to simulate stick-slip in a Couette cell and compared it with a theoretical model. Although they call it granular, the simulations represent a thin lubricat ion layer rather than bulk solids.

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82 Experimental Setup In this chapter two types of testers are used, the Schulze cell and the Uniaxial cell. In Chapter 2 the Schulze cell is described in detail. The signal from the Schulze load cells was recorded digitally using a Newport IN FCS-001B amplifier and a LabJack U12 data acquisition device. The experiment al setup for the Uniaxial tests is described in chapter 4. In this chapter the stick-slip da ta that are presented in chapter 4 are investigated in depth. Two powders that show stickslip are used in this resea rch. For its wide use in industry and easy availability, cornstarch has been chosen. The starch particles are somewhat spherical with flattened sections as can be seen in Figure 5-4. The mean particle size is 13 m with a standard deviation of 7 m. The second powder that is chosen is a polystyrene powder. The powder is produced by Kodac Company and is used as a milling medium for pigment milling. The po lystyrene powder is available in several particle size distributions. Th e two different sizes used have a mean particle size of 9 m and 40 m with a standard de viation of 6 and 9 m, respectively. As can be seen in Figure 5-4 the polystyrene pa rticles are very spherical. That makes the powder a good model powder for DEM simulations. Figure 5-4: SEM images of cornstarch (l eft) and polystyrene particles (right).

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83 In the Schulze tester the angular velocity of the base can be adjusted between zero and 0.025 RPM. The linear shear velocity in the angular cell is not constant throughout the sample. At the outside of the sample the linear velocity is tw ice as large as at the inside. To calculate a repr esentative (overall) velocity, the weighted velocity v is reported as calculated with equation 5-2. In this equation Ri and Ro are the inner and outer diameter of the sample and is the angular velocity. 2 2 3 33 4i o i oR R R R v 5-2 For the experiments where the moisture content of the powder was varied, the moisture content was determined using an Ohaus MB 45 Moisture Analyzer. To reach a required moisture level the moisture conten t was measured and an additional amount of water calculated and added. The powder w as mixed for 20 minutes using a conventional kitchen blender. The required moisture leve l was checked using th e moisture analyzer. Results of Schulze Tests Figure 5-5 shows a typical stick-slip si gnal from a Schulze Shear Cell. During the rise of the shear stress, the stick phase, the powder does not sh ear. When it reaches a maximum stress, it slips or shears quickly until a minimum stress value is reached. The stick period is long compared to the slip period. It can be seen in th e figure that, after a short initial period, both the periodicity a nd the maxima and minima are constant for a certain experiment. Therefore the signal can be called a periodic steady state. The maxima correspond to what is traditionally considered the steady state value. To verify that there is no powder de formation during a stick period, a linear displacement potentiometer was a ttached to the top lid to measure the lid displacement (see Figure 5-6). The Omega LP 804 potentiometer is attached to the lid through a slot

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84 which enables the lid to m ove up or down with the powder. The potentiometer has a linearity of 1% (full scal e), a repeatability of 12 m, a hysteresis of 25 m, and an incremental sensitivity of 0.13 m. 0 0.5 1 1.5 8910111213 Time (s)Shear Stress (kPa) stick slip maxima minima Figure 5-5: Typical stick-sl ip signal from a Schulze cell. Figure 5-6: Picture of th e potentiometer fixture to the Schulze cell lid. During an experiment the base, which cont ains the powder samp le, rotates with a constant angular velocity. If the sample does not deform during the stick phase the top lid will move with the powder and there will be a deformation of the most flexible parts in

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85 the system. Since the stick-slip events ha ve a constant maximu m and minimum shear stress and period, this deformation is an elastic deformation. Appendix C shows the stiffness calculation for the system of base, pow der, top lid, tension bars, and load cells. The base of the tester and the t op lid are very rigid. The load cells and tension bars are the most flexible but account for only 50% of the el astic deformation. The rest of the elastic deformation is within the powde r. Therefore, if the displace ment is measured accurately, stick-slip can be used to measure th e bulk elastic modulus for a powder. Figure 5-7 shows the horizontal top lid di splacement (at the potentiometer point) together with the shear stress. Since the horizontal top lid displacement is measured relative to the tester table, the displaceme nt oscillates between a maximum and minimum value. The absolute value of these displacements is of no significance, only the relative. The vertical lines indicate the slips of th e powder, while the jagged curved sections indicate the stick phases. It can be seen that the stress drops as soon as the top lid slips. In Figure 5-8 the horizontal displacement data ar e corrected for the rotation of the base, making the base with the powder sample the fram e of reference. It can be seen that the top lid has a clear stepped move ment with long periods of s tick, the horizontal sections, and very short slips, the vertical sections. A slight deviation from th e horizontal plane is noticeable immediately after the slip and just be fore the next slip, indicated in the Figures with an A and C, respectively. This would m ean that there is relative motion between the top lid and the base with sample during the stick phase. At the end of the stick phase (section C) this is due to creep of the powder. It can be seen in Figure 5-7 that the slope of the deformation changes, due to the depe ndency of creep on the increasing stress. The movement immediately after the slip is a shea r at the rate of the tester. The horizontal

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86 section A in Figure 5-7 indicates th at there is no movement of th e lid relative to the table, thus the powder deforms at the rate of the base. 1.35 1.375 1.4 3030.531 Time (s)Displacement (mm)3.9 4.8 5.7Shear Stress (kPa) C B A D Figure 5-7: Typical signals for the horizontal top lid displacement (grey, left axis) and shear stress (black, right axis) du ring stick slip in a Schulze cell. 5.2 5.3 5.4 3030.531 Time (s)Deformation (mm) A B D C Figure 5-8: Typical horizontal displacement of the top lid relativ e to the base during stick-slip in a Schulze cell. To summarize, a stick-slip event can be divided in four sections; A stick phase where the powder be haves rigidly/elastically (B) A phase where the powder creeps (C) A very fast slip (plastic deformation) (D) A slow phase where the powder de forms plastically (A)

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87 A similar breakdown was given by Cain et al. (2001) although they did not report an extended plastic deformation (A). Influence of Normal Stress The normal stress applied to the powder is varied between near zero and 10 kPa, which is the recommended limit for the tester. Both the starch powder and the polystyrene 40 m powder are investigated. The powder is sheared and when the maxima and minima are constant the stic k-slip signal is recorded for ten to fifty events. The mean of the maxima and the mean of the minima of the stick-slip signa l are calculated. These maxima and minima are plotted in Figure 59 as a function of the applied normal stress, together with the magnitude of the stick-slip events. 0 2 4 6 0246810 Normal Stress (kPa)Shear Stress (kPa) Maxima Minima Magnitude Figure 5-9: Maxima, minima, a nd magnitude of stick-slip as a function of normal stress for the 40 m polystyrene powder with linear trend lines, measured in a Schulze cell.

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88 The magnitude is the difference between the maximum and minimum value, which is the drop in shear stress during a slip ev ent. The data shown are for the polystyrene powder and the magnitude of the slip is 25% of the maximum shear stress. The starch powder shows a similar result All three parameters increase linearly with increasing normal stress, following a Coulomb-type behavi or. This supports th e conclusion that the measured stick-slip signal is a periodic stea dy state. Extrapolation to a zero normal stress level indicates that there is no apparent stick-slip wh en there is no normal load. 0 5 10 15 20 0 5 10 15 20 Frequency (Hz)Power 4.0293 Hz Figure 5-10: Power plot from a Fast Fourier Transform of a low stress experiment (left) and a high stress experiment (right). The data acquisition system sampled 300 shear stress points per second, enabling the measurement of a frequency up to 150 Hz. The frequency of the stick-slip signal was determined using a Fast Four ier Transform technique in MA TLAB. The power plot from the Fast Fourier Transform showed a distinct frequency peak, whic h corresponds with the frequency that is determined manually. Figur e 5-10 shows a typical power plot for a low stress and a high stress experiment. For low stresses several peaks are noticeable due to noise in the signal. For higher stresses the si gnal to noise ratio is high, resulting in one frequency peak.

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89 0 3 6 9 12 15 18 0246810 Normal Stress (kPa)Stick-Slip Frequency (Hz) nf 1 = 0.073 s/kPa = 0.056 s Figure 5-11: Frequency of stick-slip f as a function of the applied normal stress n for 9 m polystyrene powder. The data points are measured with a Schulze cell and the curve is a fit of the equation shown. The frequency of the stick-slip signal decreases with increas ing normal stress as shown in Figure 5-11. The data shown are for the polystyrene 40 m powder, but all the powders investigated show th e same behavior. The data are used to fit equation 5-3, where is a stress coefficient and is a time constant. The equation indicates that the inverse of the frequency, which is the time scale, increases linearly with the normal stress. nf1 5-3 Influence of Shear Velocity Several researchers [Bucklin, Schwedes, etc] have investigated the effect of the shear velocity on stick-slip and found th at the frequency increases with increasing

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90 velocity. The polystyrene powde r shows the same behavior as shown in Figure 5-12. The velocity as defined by equation 5-2 is vari ed from 0.05 to 0.13 mm/s, which is the maximum velocity of the Schulze tester. A crit ical velocity as defi ned by Bucklin et al. could not be defined since such a velocity is higher than th e tester capabilities. 0 2 4 6 8 10 0246810 Normal Stress (kPa)Stick-Slip Frequency (Hz) 0.13 mm/s 0.09 mm/s 0.05 mm/s Figure 5-12: Influence of shear veloci ty on the stick-slip frequency of 9 m polystyrene powder measured in the Sc hulze cell. Equation 5-3 is used for the curves. This dissertation is conc erned with the length scales in powders. The frequency f can be re-plotted in a length scale l using the applied shear velocity v as shown in equation 5-4. The frequency curves from Fi gure 5-12 collapse onto one characteristic length curve, as shown in Figure 5-13. Therefore, the frequency or time scale of stick slip events is a tester artifact, depending on the shear velocity. The characteristic parameter that defines stick-slip is the length scale. The length scale increases approximately linearly with the applied stress for the st ress range measured. Extrapolation of the

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91 characteristic length to a zer o stress level gives an inter cept with the y-axis of 10 m which is approximately one particle diameter. This would mean that stick-slip correlates the macroscopic scale, the shear stress, with the microscopic scale, the particle size. f v l 5-4 0 20 40 60 80 100 0246810 Normal Stress (kPa)Stick-slip Length Scale (mm) 0.13 mm/s 0.09 mm/s 0.05 mm/s Figure 5-13: Stick-slip length scale of 9 m polystyrene powder as a function of normal stress for three different shear velocities of the Schulze cell. Influence of Particle Size To investigate the correla tion between the char acteristic stick-slip length and the particle size, the experiment was repeated with the 40 m polystyrene powder. The 40 m powder also shows an incr ease of the length scale with increasing normal stress. The increase is about twice as la rge compared with the 9 m powder. The results are not exactly linear but slightly curv ed. Extrapolation of the data to high stresses could indicate an asymptotic maximum length scale. W ithin the data range measured a linear

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92 approximation is acceptable. Extrapolation of the data to a zero stress level gives a characteristic length scale of just under one particle diameter. In Figure 5-14 the characteristic stick-slip lengths of both pow ders are presented as a function of the normal stress. The length scale is reported relative to the particle size, in number of particle diameters, instead of the absolute size in millimeters. Both powders have a characteristic length at a zero stress level of about one particle diamet er. When the normal stress is increased, the relative characteristic length in creases to several particle diameters. This means that the number of particles involved in a stick-slip event incr eases with stress. It is proposed that a slip is a movement of cl usters of particles rather than individual particles. At low stresses th ese clusters are single particles, but for increasing stress the clusters consist of several particles. 0 2 4 6 8 10 0246810 Normal Stress (kPa)Length Scale (Number of Particles) 0.13 mm/s 0.09 mm/s 0.05 mm/s 0.13 mm/s 9 m polystyrene powder 40 m polystyrene powder Figure 5-14: Characteristic length scale of stic k-slip in number of particle diameters as a function of the normal stress for th e 9 and 40 m polystyrene powder, measured with the Schulze cell.

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93 As said, the increase of the absolute ch aracteristic length with normal stress is twice as large for the 40 m powder relative to the 9 m powder. As can be seen in Figure 5-14, the relative characteristic le ngth of the 40 m powder increases only half that of the 9 m powder. This means that at a certain stress level clusters in the 40 m powder contain fewer particles th an the clusters in the 9 m powder. This indicates that cohesion is a key parameter in the formation of the clusters. A powder consisting of small particles is more cohesive, t hus the clusters will contain mo re particles relative to a powder with larger particles. Influence of Moisture Content The cohesion of a powder is not only depe ndent on the particle size. As discussed in chapter 3, the cohesion can be manipulat ed by adding small amounts of oil or water. To investigate the influence of the cohesion in this work, small amounts of water were added to the 40 m powder. The result on the cohesion is show n in Figure 3-2. Interestingly, the powder did not exhibit stic k-slip above a moisture content of 0.45%. Using the particle size distri bution in Appendix B the thickness of a water layer on the particles can be calculated, assuming a uniform water distribution ove r the surface of the particles. A moisture content of 0.45% corr esponds to a water layer thickness of 33 nm. The magnitude of stick-slip increases with increasing moisture content as shown in Figure 5-15. The data are linear, but extrapolat ion to a zero stress level would result in a negative magnitude for low moisture contents This is not realistic so the curves are expected to be non-linear clo se to a zero stress level. Figur e 5-16 shows the slopes of the curves from Figure 5-15 as a function of the mois ture content. The slopes seem to have a linear relationship. It is not ve ry likely that the relationship is linear outside the measured range. As said, no stick-slip was noticed for moisture contents above 0.4%. Extrapolation

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94 to a moisture content of zero percent will give a near zero magnitude of stick-slip. This is not expected to be realistic. A certain vol ume of moisture is n ecessary to from liquid bridges between the particl es and thus increases the cohe sion. Therefore a constant cohesion value and stick-slip magnitude is e xpected below a certain moisture content. 0 0.5 1 1.5 2 2.5 024681012 Normal Stress (kPa)Magnitude (kPa) 0.40% 0.33% 0.26% 0.22% moisture content: Figure 5-15: Magnitude of s tick-slip as a function of the normal load for different moisture contents in 40 m polystyre ne powder with lin ear trend lines, measured with a Schulze cell. The relative magnitude of stick-slip mag,rel, which is the magnitude mag divided by the maximum shear stress max, increases with increasing mois ture content as well. This is expected since there is a dire ct relationship between the slopes n n magd d in Figure 5-15 and the relative magnitude as shown in equation 5-5. The relative magnitude varies from 22% for a moisture content of 0.22% to 38% for a moisture content of 0.40%.

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95 n n mag n n n n mag rel magd d 1max 5-5 0.07 0.11 0.15 0.19 0.23 0.20.3250.45 Moisture Content (%)d[Magnitude] / d[Normal Stress ] (-) Figure 5-16: Slope of the trend lines in Figure 5-15 as a function of the moisture content. The characteristic length is shown in Figur e 5-17 as a function of moisture content. It can be seen that the length scale increases with the moisture content, which is consisted with the proposed cluster form ation. There is an increase in both the intercept at zero normal stress and the slope. The intercept of the curves of most of the moisture contents is smaller than one particle diameter. This would suggest that the movement of the characteristic cluster is less th an one particle diameter. A cluster of less than one particle is not possible, so the minimum cluster size is one particle. The characteristic length can be less than one particle diameter. In that case the interaction between two particles vanishes when they are separated more than that characteristic length.

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96 0 1 2 3 4 024681012 Normal Stress (kPa)Stick-slip Length (# particle diameters) 0.40% 0.33% 0.26% 0.22% moisture content: Figure 5-17: Characteristic stic k-slip length as a function of the normal stress for several moisture contents of the 40 m polystyrene powder with linear trend lines, measured in a Schulze cell. Results of Uniaxial Tests The stick-slip investigation with the uni axial tester is focused on the strain controlled experiments. As said in chapter 4, the stress controlled experiments were too violent, resulting in the lo ss of powder during the experi ment. The stick-slip signal investigated is shown in Fi gure 4-6. Figure 5-18 shows a de tail of the signal at a low stress level and a high stress le vel. It can be seen that the frequency at low stresses is higher than that at high stress es. The amplitude is higher at higher stresses than at lower stresses. To quantify the magnitude and frequenc y, the curve is divided in small data sets and for every set the standard deviation and frequency is calculat ed. The frequency is calculated with the fast Fourier transform.

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97 0 1.5 00.01Axial Strain (-)Axial Stress (kPa) 7 13 19 0.090.1 Axial Strain (-)Axial Stress (kPa) Figure 5-18: Detail of Figure 4-6 at a low stress level (left) and a high stress level (right). The results for the standard deviation ar e shown in Figure 5-19 Although not a real amplitude measure, the standard deviations tu rned out to be more suited to represent the magnitude of the signal. The standard deviat ion increases linearly wi th the axial stress. Above 15 kPa the signal seems to level off. Th e reason for this is the stepped stick-slip signal as can be seen in the inset and in Figure 5-18. The linear dependency of the magnitude with normal stress is consistent with the results from the Schulze tester. 2 4 6 8 10 12 14 16 18 20 0.2 0.4 0.6 0.8 1 1.2 Stress (kPa)Standard Deviation (kPa) Figure 5-19: Standard deviati on of stick-slip signal as a function of the axial stress.

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98 The frequency signal is shown in Figure 520 as a function of both stress and strain. Although the signal is noisy, a clear decrease of the frequency with stress can be seen. The frequency as a function of the stress shows a dependency similar to the Schulze results, i.e., equation 5-3. Th e results cannot directly be co mpared since the velocity of the uniaxial tester is larger and the stresses on the x-axis are different. For the uniaxial results the axial stress is plotted while for th e Schulze tester the normal stress is plotted. 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 Stress (kPa)Main Frequency 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 2 4 6 8 10 Strain (-)Main Frequency Figure 5-20: Frequency of s tick-slip signal as a function of the stress and strain. Mechanism of Stick-slip The proposed mechanism of stick-slip is di fferent from that generally used in the field of tribology where the friction between two surfaces is consider ed. In powder flow there are many surfaces in contact, i.e., every individual particle is in contact with several other particles. This system of particles fo rms a structure that has to be disturbed to deform the powder. When a powder exhibits st ick-slip, the powder is able to withstand the stress up to a certain value, at which the pow der slips. This slip is very violent. After a certain movement, the newly formed stru cture in the powder can withstand the now

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99 lower stress. This cycle is repeated and th e movement is constant every time during an experiment. Since the size of the movement is dependent on the cohesion of the powder it is proposed that the powder moves in cluste rs. The size of these clusters defines the movement of the powder. When the clusters are small, with low stresses and no cohesion, the clusters consist of singl e particles. When the stress or cohesion is increased the clusters consist of several par ticles and a collapse of the st ructure of clusters will occur when the powder slips. A clear distinction can be seen between stic k-slip in stress cont rolled and strain controlled experiments. This confirms the m ovement in clusters. When a powder is stress controlled it means that the stress path is defined, and the powder deformation is measured. When the stress is increased the cl uster structure will collapse once a critical stress is reached. The powder will deform w ith constant stress until a new structure is formed that can withhold this stress. This newly formed structure has its own critical stress at which it will collapse. The result is that stick-slip du ring stress controlled compaction will result in a few very violent deformations at which the powder deforms a significant distance. In a typical stress-strai n plot the slips will be horizontal lines as shown in Figure 4-8. Strain controlled experiments, as with the Schulze cell, dictate a certain deformation path, while the stress response fr om the powder is measured. Theoretically, during a strain controlled experi ment, the particles (or clusters) move at a certain rate. In practice this is not the case. A cluster of par ticles will not move until its critical stress is reached and the structure colla pses. Until the cluster moves, the stress will increase due to elastic deformation of the powder and tester system. When the cluster structure collapses,

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100 this elastic energy is released and the stress will drop. The clusters move and a new stable structure is formed. Compared with the stress controlled expe riment this new stable state is reached quickly since the stress is redu ced immediately. Therefore, during strain control the stress oscillates continuously. It should be noted though that there is no real strain control since the powder deforms in steps. The explanation above provides an explanat ion for the observed critical stick-slip velocity that was found by sever al researchers. The powder slip s with a certain velocity. When the shear velocity of the tester is larger than the slip velocity of the powder, there is no relaxation of the elastic energy in the sy stem possible so the stress does not drop. A constant stress, which is higher than the cr itical stress, will result in a continual deformation. Hence no stick-slip will be observed. This feeds the proposition that all powder s have stick-slip but not all powders exhibit it. All powders consist of particles that form clusters which define the structure.

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101 CHAPTER 6 INTER-PARTICLE FORCE MEASUREMENTS Tribology is the science of interacting surfaces. It enta ils the studies of friction, wear, and lubrication. The word trib ology, derived from the Greek word tribo meaning rubbing, is relatively new. It was introduced in the 1960’s while evidence has been found that the ancient Egyptians alre ady used lubricants. More rece ntly, due to the advancement of measuring techniques, the subfield of microand nano-tribology has evolved. In microand nano-tribology fr iction, adhesion, wear, and thin-film lubrication is investigated on a scale ranging from atomic scale to micro scale. The measurement of forces between particles in powders bel ongs to this subfield of tribology. Several methods are available to perfor m microand nano-tribology measurements. The first method, the scanning tunneling mi croscope (STM) was developed by professor Binning and his group in 1981 (Binnig, 1992; Binnig and Rohrer, 1983; Binnig et al., 1982). In 1986 Binning and Rohrer received the Nobel Prize in physics for their work. Binning’s group continued to develop the At omic Force Microscope (AFM), now the most widely used measurement technique (B innig et al., 1987; Bi nnig et al., 1986). The general name for STM’s and AFM’s is scanni ng probe microscopes (SPM’s). SPM’s are very versatile and can generally measu re surface and adhesion forces, measure mechanical material properties, and image surfaces with a sub-nanometer resolution. An extensive description of the different SPM measuring techniques can be found in the Handbook of Micro/Nano Tribology (Bhushan, 1999). The technique used in this work is the AFM.

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102 In this chapter the interaction between polystyrene particles is measured. The goal is to measure the inter-particle forces, adh esion or pull-off force, friction, and particle stiffness. This information is incorporated into a discrete element mode l at the end of this chapter to simulate the stick-slip behavior that was observed in the previous chapter. Atomic Force Microscope A short description of the Atomic For ce Microscope (AFM) technique is given below. An extensive description of the AFM, including measurement options not used in this work, can be found in the Handbook of Micro/Nano Tribology (Bhushan, 1999) or the dissertation of Meurk (2000). The atomic force microscope consists roughly of five components as presented in figure 6-1. Controller Piezoelectric cr y stal Laser Detector Cantilever Sample Figure 6-1: Schematic of an Atomic Force Microscope. The sample surface is placed on a piezoelec tric crystal which can control the position of the sample in the x-, y-, and z-direction with na nometer precision. A cantilever with a sharp tip touches the sample and scans the surface when the piezo moves the sample around. A laser beam is proj ected on the backside of the cantilever tip and reflected onto a position se nsitive photodiode detector. Du ring the scan, the asperities on the surface will bend the cantilever up or down, causing the laser reflection to shift on

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103 the detector. This shift is correlated to the height of the asperiti es, enabling the mapping of the surface. Some AFM’s use an inverted system, wher e the substrate is on a rigid table, while the cantilever is attached to a piezoelectric cr ystal. The advantage of that setup is that the substrate can be of a much larger size. As the name indicates, an AFM can also be used to measure the force between the cantilever tip and the surface. Two types of force measurements are possible, normal force and lateral force measurements. The normal force measurement is used to measure the interaction between the tip and the sample when the tip is appro aching, touching, and retracting. The lateral force measurement is used to measure the fr iction between the tip and the sample surface when the tip is moved over the su rface laterally. N eutral p osition Figure 6-2: Schematic of lateral force m easurement with the AF M showing torsional deflection or twisting of a rectangular cantilever. The normal force interaction is calculate d from the vertical deflection of the cantilever during the measurements. The lateral force interaction is calculated from the torsional deflection or twist of the cantilever as illustrated in Figure 6-2. For small deflections the cantilever follows Hooke’s law (see equation 61) where the force Fi is the product of the deflection i and the spring constant or stiffness ki. The index i indicates either a normal (c) or torsional () measurement. In the past researchers used

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104 the same k value for both torsional and normal deflection measurements. This is not correct since the two cons tants rarely coincide. i i ik F 6-1 The normal spring constant kc can be calculated using equation 6-2, which is derived from the mechanical theory for a bendi ng body that is rigidly attached at one end. In this equation E is Young’s modulus, l the cantilever length and I the moment of inertia. The shape of the cantilever defines the moment of inertia. Cantilever s are generally either v-shaped or rectangular-shaped. For a rectangular cantilever (straight beam) I is equal to 3 12 1wt with w the width of the beam and t the thickness. For a v-shaped cantilever the determination of I is not that straightfo rward. In both cases the kc value depends on the cantilever thickness cubed. The cantilever thickness is generally between 2 and 6 m and it is difficult to measure it accurately. Theref ore a method is used to define the spring constant experimentally. The eigenfrequency f of a spring is given by equation 6-3, with meff the effective mass and k the spring constant. The effective mass depends on the shape of the cantilever, including the tip and is thus hard to define. When a mass madd, that is significantly larger than the effective mass of the spring, is added to spring, the spring constant of the spring can be approximate d well with equation 6-4. The AFM can oscillate the cantilever w ith a frequency up to several megahertz to find the eigenfrequency. 33 l EI kc 6-2 effm k f2 1 6-3

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105 add cm f k2 24 6-4 To measure translational fo rces rectangular cantilevers are more suitable than Vshaped cantilevers. The torsional spring constant k can be calculated using equation 6-5, with G the shear modulus, w the cantilever width t the cantilever thickness, h the tip height, and l the cantilever length. When the normal stiffness is determined from the frequency shift, the torsional stiffness can be calculated by combining equation 6-2 and 6-5, which yields e quation 6-6, where is the Poisson ratio. An alternative method to measure the torsional spring constant was proposed by Bogdanovic et al. (2000) but this method is quite laborious. l h Gwt k2 33 6-5 ) 1 ( 3 22 2 h l k kc 6-6 The AFM can measure both short range a nd long range forces. Figure 2-10 shows typical force interaction curv es. The interaction can includ e electrostatic and van der Waals forces, which can be attractive or repu lsive, and adhesion or pull-off forces. Other interaction forces, such as capi llary and electromagnetic for ces are not investigated with the AFM in this work. Measurements usuall y show a different curve for extension (approaching) and retraction due to the pull-off force. Figur e 6-3 shows th e approaching and retraction curves as measured with th e AFM for a system with and without electrostatic repulsion. The data shown are th e raw data, with the cantilever deflection on the x-axis and the displacement of the base of the tip on the x-axis. At position A the cantilever tip is far from the substrate a nd there is no interaction thus no cantilever deflection. When the tip approaches there can be an electrostatic repulsion (B), which

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106 will bend the cantilever without contact between the tip and substrate. When the tip is near the substrate, the van der Waals forces pu ll the tip to the surface, resulting in a snapon deflection (C). When the tip is in cont act, the cantilever will bend linearly in the constant compliance regime (D). A E B C D Extension curve Retraction curve A E C D Figure 6-3: Typical AFM result for force interaction measu rements between a tip and a substrate, with electrostatic repul sion (top) and w ithout (bottom). There is usually a slight off-set between the extending and retracting curve in the constant compliance regime due to hysteresis in the piezo elem ent. When the cantilever is retracting, the deflection will become nega tive due to the adhesion between the two surfaces. When the cantilever force equals th e pull-off force (at point E), the tip will snap-off and the cantilever will return to the neutral position. When the interaction between particles is required, one particle can be attached to the can tilever and another to

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107 the substrate. The above description for tip -substrate interaction during extension and retraction is valid for particle-particle inter action as well. The force curves in Figure 2-10 are shown as a function of the particle separation distance, while the curves in Figure 6-3 are a function of the z-position of the tip base. In AFM measurements distance is not a straightforward measure. Due to the roughness of the surface one can choose to take the separation to be between the ti p and either the top or bottom of an asperity, or the mean asperity size. It should be noted that the distance between the tip and a surface is not determined exact with AFM. The distance d between the tip and an asperity is determined from the cantilever deflection c, the distance between the un deflected cantilever and the sample surface z, and the surface roughness s using equation 6-7, as illustrated in Figure 6-4. s cz d 6-7 s d c z substrate y Figure 6-4: Schematic showing the relative separation betw een a cantilever tip and the substrate. For AFM force measurements the control parameter is the y position of the substrate base (or the tip base for an inverted AFM). The distance z between the sample surface and the neutral tip position is controlled by this y position. This control is a relative control though. Th e absolute value of z is not known for two reasons. Firstly, the

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108 thickness of the sample is not known and secondly, as said, the sample surface has a roughness, indicated with s, which makes it arbitrary what that exact thickness would be. Care should not only be taken when defining the separation distance. The interpretation of the force data needs special attention as well. The system of cantilever, tip, and substrate can be modeled as a combin ation of springs and dashpots interacting with a mass as shown in Figure 6.5. The cantilever with mass meff is modeled by the top spring and dashpot, simulating the energy st orage and dissipation, respectively. The bottom spring and dashpot simulate the interac tion of the cantilever with the substrate. The spring constant ki is not a constant but depends on the distance between the tip and the substrate as well as the deflection of the cantilever. For large separation distance, when all interaction forces have vanished the spring constant is zero. When the separation distance decreases, the interaction fo rces, e.g., van der Waals or electrostatic, start interacting between the tip and subs trate. The interaction forces will be in equilibrium with the cantilever deflection fo rce from equation 6-1. Therefore, for this interaction region the appa rent spring constant ki is very high. When the separation distance is zero the tip is in contact and starts indenting the substrate, which results in a spring constant ki equal to the substrate stiffness. ckcki( c,z) i z c meff Figure 6-5: Schematic of the cantilever, tip, and substrate, represented by springs and dashpots.

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109 Experimental Setup The experiments are performed with a Dimension 3100 Atomic Force Microscope from Digital Instruments. The cantilevers used are rectangular “tapping mode” silicon cantilevers, type NCH-W from Pointprobes. A polystyrene particle is glued to the cantilever using Loctite M-21H P Hysol medical grade epoxy adhesive. The particles are glued to the cantilevers with the aid of a micro stage and an Olympus BX60 Optical Microscope with SPOT RT Digital Camera. Pa rticles are glued to a smooth silicon wafer with the same adhesive to form a substrate. Fo r this, a thin layer of glue is applied to the wafer and the powder is rained on the glue. After drying, the excess powder is blown off with compressed air. Figure 6-6 shows two pict ures of a particle attached to the cantilever as well as a part of the substrate with particles. The glue is viscous to prevent coating of the particles with the glue, which is visually inspected. Figure 6-6: Pictures of part icles attached to a cantile ver (left and middle, 500x magnification) and the substrate of particles (right, 200x magnification). The eigenfrequency of the cant ilever with pa rticle is measured and equation 6-4 is used to calculate th e cantilever stiffness. The stiffness of the cantilevers is 54 N/m.

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110 Adhesion Force Measurement The pull-off force between particles is m easured to define the adhesion force. The polystyrene powder did not show any si gnificant electrostati c repulsion and the measurements look like the bottom graph in Figur e 6-3. The data are replotted to reflect the particle interaction force as a function of separation distance. Figure 6-7 shows a typical result for a 47 m particle in contact with a particle of about 80 m. The force is scaled as force over radius to exclude the particle size. AFM measurements generally show a large scatter. It is not uncommon to ha ve one or two orders of difference between measurements. Therefore a large number of measurements is necessary to acquire a statistically valid result. For this research only a limited number of 22 measurements is made. The results are quite consistent though. The mean pull-off force for 47 m particles is 1236 nN with a standard deviation of 406 nN. -100 0 100 200 300 020406080 Separation (nm) Force/Radius (mN/m) extend retract pull-off force Figure 6-7: Typical force-displacemen t result for polystyrene particles.

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111 Due to the small number of measurement s no reasonable pull-off force distribution can be plotted. Note that due to the high value of the pull-off force and the quick recoil of the cantilever only one data point is measu red during the flight of the cantilever. Therefore, no conclusions can be drawn from the data about the in teraction forces after pull-off. When particles deform plastic ally or visco-elastically during loading, the pull-off force will increase with increasing maximum loading force due to the increasing (semi) permanent contact area. The polystyrene partic les did not show such behavior within the force range that was used. Therefore it can be concluded that the polystyrene particles deform elastically at contact during the AFM measurements. The pull-off force is due to the van der Waals attraction between the particles. Using equation 2-15 the distance between the pa rticles at contact can be calculated. The theoretical Hamaker constant for polystyrene is 6.6*10-20 J which has been confirmed experimentally (Israelachvili, 1985). The resu lts are shown in Table 6-1 for different van der Waals force values. The values are close to the minimum Born separation distance of 2 to 3 A which is to be expected si nce the particles are relatively smooth. Table 6-1: Calculated particle se paration distance between a 47 and 80 m polystyrene particle for different van der Waals forces. van der Waals Force Value (nN) Particle Separation (A) Mean minus one SD 830 4.4 Mean 1236 3.6 Mean plus one SD 1642 3.1 Friction Force Measurement The lateral or friction force measurement is conducted by moving the cantilever laterally back and forth over the substrate as de picted in Figure6-2. When a particle is

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112 attached to the cantilever and another to the substrate, the friction between the particles can be measured. A typical friction loop is s hown in Figure 6-8. The plot shows the raw data, the torsional deflection of the cantilever in units of voltage on the detector on the yaxis and time on the x-axis. The time reverses for the retrace. The scanning frequency and distance are known so the time can easily be converted in a length scale. trace retrace A 2* Ffriction Figure 6-8: Typical AFM fr iction loop of a measurement between two polystyrene particles. The friction loop can be interpreted as fo llows. At point A the cantilever starts moving to the right, but the particle that is attached to it cannot slide due to the interparticle friction. This causes the cantilever to bend torsionally while the particle rolls. From the slope of the curve at point A the sen sitivity of the detector can be calculated. This sensitivity is used to convert the voltage into a de flection which, using equation 6-1, can be converted into a lateral force. When the lateral force equals the (static) friction of the particles, the partic le starts to slide over the surface of the other particle. At this point the particle does not roll anym ore and the lateral force stays constant. At the end of the trace the cantilever slows down, stops, and reve rses its direction. The particle will stop sliding, roll in the other di rection and start sliding when the critical lateral force is

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113 reached. This direction is calle d the retrace. As shown in Fi gure 6-8, the friction force is half the distance between the two horizontal sections. The external normal force on the particle can be controlled, enabling the construction of a friction curve. For some material the friction is more or less dependent on the sliding velocity. No such dependency could be observed for the friction force between the polystyrene particles. Interestingly, while the polystyr ene particles show cl ear stick-slip at a macroscopic scale, no stick-slip could be observed during the particle friction measurements at any velocity. Other researc hers have observed stick-slip between particles. Meurk (2000) reported a critical velocity for coated silica of 3.2 m/s below which stick-slip occurs. Even at velocities of 0.2 m/s no stick-slip was observed for polystyrene. 0 6000 12000 -6000-4000-200002000400060008000100001200014000 Normal Force (nN)Friction Force (nN) Figure 6-9: Friction force as a function of the external normal force for polystyrene particles. The results of tw o identical tests are shown, with connecting lines to guide the eye. The arrows indicate the direction of the experiments.

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114 The results for the friction experiment s are shown in Figure 6-9. The scanning frequency was 3 Hz with a scan width of 1 m and thus a scan velocity of about 6 m/s. The friction is evaluated on two different positions and it can be seen that the results are very reproducible. During an experiment, th e normal force is stepwise increased to a maximum value and subsequently stepwise decreased. At every normal force level the friction loop as shown in Fi gure 6-8 is captured and the friction force evaluated. At the start of an experiment, at a norma l force slightly above zero, the friction force immediately jumps to a high value. When the normal force is subsequently increased the friction force seems to incre ase near linearly. When the normal force is reduced, the friction force does not show hysteresis until the zero load condition is reached. For a negative normal force, the particles stay connected and show a positive friction. The friction becomes zero at a normal force of about –1460 nN. The reason for this hysteresis is the for ces between the particles themselves. The normal force presented in the graph is the exte rnal force that presses th e particles together while, as shown earlier in this chapter, ther e is also an attractive force between the particles themselves. The absolute value at which the friction becomes zero is about half a standard deviation higher than the measured mean pull-o ff force. In Figure 6-10 the measurements are re-plotted usi ng the total normal force, which is the external force plus the pull-off force which is chosen to be 1460 nN. As discussed in chapter 2, several models are available for frictional behavior. The data clearly show that the friction force betw een the particles is not a linear function of the normal force for low normal forces. The data are used to fit four models using a least squares method. The well-used empirical power model (equation 2-26) fits the data well

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115 and gives a power of 0.45. This value is lower than the gene rally accepted values between 1 and 2/3. The mode l proposed by Adams et al. ( 1987) (equation 2-27), fits the data very well but has a negative coeffici ent of pressure. A nega tive coefficient of pressure means that for large normal for ce values the friction force decreases with increasing normal force, which is unrealis tic. Equation 2-25 has been evaluated using both the DMT and JKR model. It can be seen th at the JKR model giv es a better fit than the DMT model. The JKR model can not be evaluated for very small forces. 0 3000 6000 9000 12000 0350070001050014000 Normal Force (nN)Friction Force (nN) Empirical DMT JKR AdamsF A F A F A F F FDMT friction JKR friction DMT friction friction 87 0 10 27 : Adams 10 14 : JKR 10 16 : DMT 163 : Empirical3 3 3 45 0 Figure 6-10: Friction force as a function of the total normal force for polystyrene particles. The curves show the best least squares fit for different models; the empirical model shown in equation 226 (red solid curve), the DMT model (green dotted curve), the JK R model (black solid curve), and the Adams et al. (1987) model (blue dashed curve). Macroscopic friction measurements usually produce a linear de pendency of friction on load, resulting in a constant coefficient of friction. For polystyrene the reported macroscopic value is 0.5. A linearization of the measured data at normal force values

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116 above 5000 nN gives a slope of 0.45 in that re gion. It might be th at the friction curve becomes linear for higher stress values as the m odel of Adam et al. predicts. With the available cantilevers it is not possible to measu re at higher force levels. If the friction curve becomes linear at higher loads it expl ains why macroscopic friction measurements between surfaces are linear. The non-linearity is not measured with macroscopic experiments because it will only occur at very lo w force values and this will be within the error of the measurement. For the modeling of powders with DEM t echniques this non-lin earity cannot be ignored. The use of a macroscopically m easured friction coefficient will largely underestimate the friction force between the particles. This underestimation will be compounded in the final results of the DEM si mulation. Both the inherent van der Waals forces and the non-linear frictiona l behavior need to be included to yield realistic results. Inter-particle Force Modeling The simulations shown in chapter 3 and 4 do not incorporate inter-particle forces other than the contact mechanics. In the Itasca PFC 2D DEM code extra forces can be assigned to particles as body forces. This feature is used in the following simulations to simulate van der Waals forces and thus adva nce towards more accurate simulations. In the Itasca code contacts between particles are detected befo re there is physical contact between the particles. This en ables the introduction of the van der Waals attractive forces before the particles actually touc h. The drawback of the code is that the distance at which virtual contacts are detected cannot be contro lled. Therefore, care ha s to be taken using this feature. The measurement of the pull-off force of polystyrene particles show good agreement with the theoretical van der Waal s force as shown in Table 6-1. From the

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117 friction experiments it can be seen that JKR m echanics predicts the data better than DMT mechanics. The Itasca code does not suppor t JKR mechanics though, so the DMT model is chosen for the simulations (equation s 2-15 and 2-21). The Itasca code supports Hertzian contact mechanics and the adde d van der Waals forces complete the DMT model. The maximum van der Waals force is set at the value corresponding to a separation distance of 3 A. This is necessary since during the simulation the particles can actually touch and overlap which would re sult in unrealistic van der Waals forces. The Hamaker constant for polystryrene is chosen to be 6.6*10-20 J (Israelachvili, 1985). No electrostatic or capillary forces are included. Atomic Force Measurement Simulation To confirm the accuracy of the Itasca simulations w ith the newly added interparticle forces the AFM measure ments are simulated. In the Itasca code a row of particles can be glued together using the parallel bond option. This bond acts as a cemented collar, enabling the transfer of a moment between th e particles. Such a row of particles can simulate the bending of a beam ve ry accurately (Itasca Manual, 2002). A row of 14 particles with parallel bonds is used to simulate a cantilever as shown in Figure 6-11. The x-position and rotation of particle 1 are fixed at zero, while the particle is moved with a constant y-velocity of 3 m/s, equal to the real AFM velocity. The radius of the particles in the beam is chosen to be 10 m, which is between the width and thickness of the actual cantilever of 33 and 4 m, respectively. The parallel bond and particle properties are chosen such that the cantilever has a stiffness of 54 N/m, equal to the cantilevers used in the AFM. The deflecti on of the cantilever is the difference in y-

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118 position between particle 1 and 14. Simulations confirmed that the particle beam has a constant stiffness for the full range of measurements (and well beyond). 200 100 14 13 12 11 10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 9 8 7 6 5 4 3 2 1 x y Figure 6-11: Discrete element model of the AFM cantilever with attached polystyrene particle and a polystyrene substrate particle. Two particles, simulating polystyrene particles, are included in the simulation. One particle (number 100) is attached to particle 14 with a parallel bond simulating the glue, while the other particle (number 200) has a fixed xand y-position right under particle 100. The rotation of particle 200 is also fixed a nd set at zero. The displacement of particle 1 is set to a cycle of alternating downwar d and upward movement. The result is shown in Figure 6-12 and the similarity with Figure 6-3 is apparent. At point A the particles are close enough th at the van der Waal s force pulls them together. Region B shows the constant complia nce of the cantilever hence the constant slope. There is no off-set between the exte nding and retracting curv e since there is no hysteresis in the piezo element in the simulati on. A straight line is drawn parallel to the constant compliance region to show that for ne gative external loading of the particles the compliance is not exactly linear. This is due to the Hertzian contact model and reflects reality. As said, a maximum van der Waals force is programmed in, which defines the value of the pull-off force at point C. When the particle snaps off, the cantilever recoils

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119 and shows a short high-frequency oscillation, resulting in point D. At point E there is no interaction between the particles. The resulting pull-off force of 73 mN/m is slightly higher than the value measured with the AFM. The mean value measured w ith the AFM is 53 mN/m with a standard deviation of 17 mN/m. The reason for this is that in the simulation the closest distance between the particles (due to Born repulsion) is set at 3.0 A. This is slightly lower than the repulsion distance values calculated from the AFM results (see Table 5-1), causing a slightly higher maximum van der Waals force. -100 -50 0 50 100 525456585105 Distance (nm)Force/Radius (mN/m) A C B D E Figure 6-12: Discrete element simulation of a pull-off AFM experiment using the DMT model. Friction Modeling For the simulation of the AFM friction m easurements three particles are used as shown in Figure 6-13. The simulation is a front view with the top particle simulating the

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120 cantilever. The middle particle is a polystyr ene particle and is glued to the cantilever particle with a parallel bond. The third particle is the polys tyrene substrate particle and its position and rotation are fixed. The x-velocity of the cantilev er particle is controlled, while its rotation is fixed at zero. The cantileve r particle is moved b ack and forth laterally to simulate the friction loop. The difference in x-distance of the can tilever particle and middle particle represents the torsional deflection of the cantilever. The deflection in the simulation is compared with the friction betw een the particles that is calculated by the code and the two results coincide. Unfortunate ly, it is not possible to include a non-linear friction coefficient in the code. Therefor e, no extensive friction simulations are performed and a friction loop is si mulated for one force level only. -2000 -1000 0 1000 2000 -0.07-0.05-0.03-0.010.010.030.050.07 Lateral Displacement ( m)Friction Force (nN) A D C E x y Figure 6-13: Friction loop measured with a DEM simulation, simulating polystyrene particles with a Hertzian contact model.

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121 The friction between the partic les is simulated without an external force, i.e., the particles are held together by the van der Waals force, whil e the Hertzian contact model is used. In Figure 6-13 the simulation starts at the top of the substrate particle. The cantilever particle is moved to the left, but due to the inter-particle friction the attached polystyrene particle does not move laterally but rolls. The slope of the curve starting at A is the stiffness of the parallel bond and repres ents the torsional spring constant of the cantilever-with-particle system. At point C th e maximum friction force is reached and the middle particle starts to slide. At point D the direction of the velocity of the cantilever is reversed. No acceleration is in cluded in the simulations, so the velocities in either direction are reached instantaneously. The polysty rene particle stops sl iding and starts to roll towards the right. At point E the maximum inter-particle friction is reached and the particle starts to slide to the right. Th e sliding is taken past the top and the same movement is achieved at the right side of the plot, co mpleting the friction loop. The resulting friction force of 1500 nN without an external normal force is lower than the measured value of 4000 nN at zero load (see Figure 6-9). The reason for this is the linear friction model that is used in the simulation. A linear fric tion model will not be able to predict the non-linear friction that is measured. Although the value of the lateral force in the simulation is off, the added va n der Waals force causes the friction to be nonzero at zero external load, which is what is measured with the AFM. During friction measurements it is important to keep the loading constant. The AFM controls the loading by monitoring the vertical cantilever deflection. When scanning a particle over another particle, th e height of the surface changes due to the curvature of the substrate particle. The AFM can correct for this but care should be taken

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122 to position the particles exactly above each other. Also, the sca nning width should be as short as possible. Figure 6-14 shows the y-position of the can tilever particle as a function of the scanning position during the simulation. -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 -0.07-0.05-0.03-0.010.010.030.050.07 Lateral Displacement ( m)Vertical Deflection (nm) A D E C B Figure 6-14: Vertical cantilev er deflection during a DEM si mulation of a friction loop. Note that the scales on the axes are thr ee orders of magnitude different. For the inset the x-axis is about 11 nm wi de while the y-axis is about 0.0002 nm high. In Figure 6-14, at point A th e simulation starts and the particle attached to the cantilever snaps on to the other particle due to the van der W aals interaction. At point B the cantilever starts to move to the left, cau sing the cantilever to go up slightly. This is due to the rolling of the polysty rene particle and the stretch of the glue, th e parallel bond. When the cantilever reaches poi nt C the polystyrene particle starts sliding and due to the curvature of the substrate particle it moves downward. At point D the direction of the velocity of the cantilever is reversed and the cantilever rises due to the rolling action of

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123 the polystyrene particle. At point E the maximum friction force is reached and the particle starts to slide. When the cantilever passes the top of the substrate particle it still ascends and does not descend until a few nanometers after the t op. This hysteresis is due to the fact that the cantilever drags the polys tyrene particle over the surface, i.e., the polystyrene particle is sli ghtly behind the cantilever. Stick-slip Modeling Theoretically, discrete element simulations should give correct macroscopic results if the input parameters are correct and the num ber of particles in the simulation is large enough. The van der Waals force is successfu lly included, but the nonlinear frictional behavior cannot be incorporated Therefore, an average coeffi cient of friction is included which is set at 1. This coeffi cient of friction will underestim ate the frictional force at low normal forces and overestimate it at high forces. The geometry that is simulated is the Jenike shear cell as shown in Figure 6-15. The number of particles in the simulation is 925, which is at the lower limit of a reasonable simulation. The particle size distri bution is Gausian with a mean particle size of 47 m and a standard deviation of 9 m, which is comparable to the polystyrene powder (see Appendix B). During the simulation th e walls forming the top part of the cell (in blue) are moved with a constant velocity to the right while the bottom part (in red) is stationary. The deformation velocity of th e walls is 0.2 mm/s. The forces in the xdirection in the top section are monitored a nd represent the shear force as measured with a real Jenike cell. These forces are the norma l forces on the vertical blue walls and the shear forces on the horizontal blue walls. It should be noted that the top wall in the

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124 simulation confines the particles while in a real Jenike cell the top wall applies a constant (overall) stress on the powder. Figure 6-15: Geometry of the Jen ike cell in the DEM simulation. The force as a function of the displacement of the walls is shown in Figure 6-16. There are clear drops in the st ress, similar to the real expe riments. The data points are equally spaced in time and the data point density is a few magnit udes larger during the rises in force compared with th e drops. This means that the rises are slow while the drops are very fast, indicativ e of stick-slip. The stick-slip ev ents are not as periodic as in the real experiment. The r eason for this can be that the number of particles in the simulation is relatively small. Also, the normal force on the top of the powder be d is not constant as with a real experiment. The powder is confined by the top wall, but the normal force on that wall is not controlled. It is generally difficult to compare 2D simulations with real experiments quantitatively. The results of the simulati on are given in total force on the walls. To calculate the corresponding wall stresses, a th ickness has to be assumed for the 2D crosssection. For this the mean particle thickness can be used as an estimate. The length of the top lid is 2500 m, with a mean particle diameter of 47 m, giving a cross sectional area of 1.175*10-7 m2. The following stresses are calculated using this cross-sectional area. The results show a stepped stick-slip si gnal (see Figure 6-16). During these stickslip events the normal stress is approximatel y 4.3 kPa. The magnitude of the major slips

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125 is 0.34 kPa with a maximum shear stress of 2.4 kPa, resulting in a relative magnitude of 14%. It can be seen in Figure 5-15 that the ma gnitude is similar to the value that is measured for the polystyrene powder with 0.22% and thus predicts the real experiments well. The relative magnitude of 14% is lower than the experimentally measured value of 22%, but is within a reasonable range. 0 0.05 0.1 0.15 0.2 0.25 0.3 00.020.040.060.080.10.12 Displacement (mm)Shear Force (mN) Figure 6-16: Shear force as a function of di splacement during a Jenike cell simulation. The length scale of the major stick-slip events in the simulation is about 17 nm. This is 1/3 of the mean particle diameter. It can be seen in Figure 5-17 that that length is slightly lower than the experimental result s but within 30% of the polystyrene powder with 0.22% water. It should be noted that the simulation is performed for a dry powder, so the resulting stick-slip lengt h is expected to be lower. To find out what causes stick-slip, the van der Waals forces or the friction, two simulations are performed without van der Waal s forces, one with a friction coefficient of

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126 0.3 and one with a friction coefficient of 1. The results are shown in Figure 6-17 together with the initial results. The simulation w ithout van der Waals forces with a friction coefficient of 1 show a similar profile as th e simulation with van der Waals forces. The maximum shear force and the magnitude are sli ghtly lower, what is to be expected. The van der Waals forces create a cohesion in the powder ensemble, giving it a greater strength. 0 0.06 0.12 0.18 0.24 0.3 00.020.040.060.080.10.12 Displacement (mm)Shear Force (mN)Fvdw = 0 = 0.3 Fvdw = 0 = 1.0 Fvdw = 0 = 1.0 Figure 6-17: Comparison of shear experiment simulations. With van der Waals force and friction coefficient 1 (gre y), and without van der W aals force with friction coefficient 1 (blue) and 0.3 (red). When there is no cohesion and a moderate friction coefficient of 0.3 in the simulation, the shear force does not significan tly rise above the x-axis. Figure 6-18 shows this result on a different scale, with an inse t of the first 0.03 mm for all three simulations. No significant increase in shear force is found for the low friction coefficient. The scatter is due to individual partic les bouncing into the walls a nd is of the same order of

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127 magnitude as with the other experiments. The scatter is no stick-s lip as found in real experiments since the timescale of the ri se and drop of the force is the same. 0 50 100 150 200 250 300 350 400 450 00.020.040.060.080.10.12 Displacement (mm)Shear Force (nN) 00.010.020.03 Fvdw= 0 = 0.3 Fvdw= 0 = 1.0 Fvdw = 0 = 1.0 0 400 Figure 6-18: Shear force as a function of the displacement with a friction coefficient of 0.3. The inset shows the first 0.03 mm of shear to compare the granular scatter between the three experiments. The grey curve includes cohesion and a friction coefficient of 1, the blue curve has no cohesion and has a friction coefficient of 1, and the red curve has no cohesion and a friction coefficient of 0.3. The results indicate that stick-slip is caused by the high fr iction between the particles. This friction enables the particl es to build stable structures, which collapse when the force becomes too large. The cohesi on aids in the formation of this stable structure. It forms an attrac tive force between the particles, pulling them together. Since all particles are preloa ded due to the van der Waals inte raction, the friction acts between the particles even at zero exte rnal force. This enables the powder to form stable structures even at low consolidation. This can be seen in Figures 6-17 and 6-18 where the shear

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128 force in the powder with c ohesion increases at a lower wa ll displacement than the two other powders. The results show stick-slip in the powder with a friction coefficient that is linear while in reality the friction coefficient is non-linear. A non-linear co efficient of friction will probably cause the powder to show a periodic stick-slip during steady-state shear. When the powder structure partly collapses, th e contact forces in the surrounding area are lowered. The non-linear friction forces will be lowered more, causing the contacts to slide at small normal force changes. A slip event will ther efore cascade through the powder until a significantly stable structure is formed. This new structure will be of the same strength as the latter structure, causing the powder to collapse at the same strength value.

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129 CHAPTER 7 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK Powder processing is very important in sev eral industries. Yet there is still a significant lack of fundament al understanding of powder be havior. Traditionally most research and design is based on trial-and-er ror, empirical models, and mechanistic models. To prevent the abundance of problems that are present in industry, design should be based on first principles. To establish models based on first principles, a bottom up approach has to be used. In powder flow th e focus has mainly been on the macroscopic scale of powders, trying to establish empirical correlations and mechanistic models. To be able to understand the fist principles of powder flow, the pa rticle interactions are key. A connection has to be made between the particle scale and the bulk powder scale. When correct inter-particle force models have been established, these can be used in discrete element method (DEM) computer simulations to simulate an ensemble of particles. The results from these DEM simulations can be used as a basis for continuous scale models. Process equipment can be desi gned using finite element method (FEM) models with the DEM results as input parameters. This research investigated the connecti on between the micro scal e of particles and the macro scale of bulk powders. Stick-slip is used to establish a characteristic length scale connecting the micro scale and the macro scale. This characteris tic length scale is called the coherence length, which is a new parameter proposed by the author. The coherence length describes how far the influence of an individual partic le reaches into the powder bulk. It is also referred to as sphere of influence of a particle. When a process is

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130 of a length scale smaller than the coherence length, the powder has to be treated as granular matter. All individual particles have to be taken into account in the design or measurement. When the scale of a process is la rger than the coherence length the powder can be treated as a bulk material and a continuum approach can be used. Conclusions When testers are used, one should always be aware of the limitations of the equipment. It is possible that tester artifacts are wrongfully interpreted as t est results. Commercially available powder shear testers have been i nvestigated often and their performance was considered known. This r esearch shows that one tester artifact, anisotropy, has been wrongfully interpreted as a powder proper ty, while in reality it is a synergistic result of powder a nd tester. It has long been known that powders have a different strength depending on the direction of measurement. This is due to anisotropy that is formed in the powder during pre-shear. Both measurements and DEM simulations show that there is a non-symmetric stress distribution in the pow der sample in shear testers. When a powder is failed in a direction other than the direction of pre-shear, this non-symmetric stress distributi on causes part of the powder to reach a new steady state deformation rather than a failu re. This will result in the observed lower failure value. The measured anisotropy is partly due to the tester geometry and is therefore called geometric anisotropy. In this work a new uniaxial tester is deve loped as a basis for the development of a true triaxial tester. The unia xial tester is designed to deform a powder either stress controlled or strain controlled, which is unique The tester shows that it is possible to form shear zones during uniaxial consolidation, while it is ge nerally considered that only compaction occurs. Different shear testers ar e available to measure powder properties.

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131 Some are stress controlled, some are strain controlled, and some are pseudo stress controlled. This research shows that the c hoice of control is not arbitrary. Depending on the powder and the application the right control mechanism should be chosen. When the concern lies in the final product strength, the fo cus should be stress c ontrol. It should be noted that it can be dangerous to use pseudo stress contro l, especially when a powder exhibits stick-slip. Stick-slip is a discontinuous or stepwi se powder flow. It causes major problems in industry since it can indu ce structural vibrations and inc onsistent flow. Other researchers have investigated stick-slip and found a dependency of the stick-slip magnitude and frequency on the shear velocity. In this work a new approach for stick-slip research is used by investigated the length of a slip event. This length shows to be independent of the shear velocity and is proposed to be a powder property. The dependency of the stick-slip length on the stress state, moisture content, and particle size is investigated. The length proved to increase with all three parameters. A relative stick-slip length is introduced, descr ibing the slip length in number of particle diameters. The experiments show that the re lative length scale decr eases with increasing particle size. The conclusion from all these result s is that the length of a stick-slip event is a function of the cohesion of the powder. In this work a novel mechanism is propo sed explaining stick-slip as a sudden collapse of the structure in the powder. For this to occur, the particles in the powder need to be able to form a semi-stable structure that will collapse when overloaded. During the collapse the energy that was st ored in the structure is re leased and the stress decreases.

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132 The collapse will last until a new semi-stable structure is formed. The formation of the structure is effected by the cohesion and stress state of the powder. The particles form a loose structure for a low cohesion and a low stress state, resulting in small collapses. The stick-slip length and magnitude will be small for these collapses. When the cohesion in creases the particles form a mo re stable structure due to the attractive forces between the particles. A higher stress state will activate the friction between the particles, enabling the formation of a stable st ructure. In both cases a larger load is necessary to break the structure. The co llapse will be more violently, resulting in a larger length and magnitude of the stick-slip. The higher stresses and cohesion will cause groups of particles to move during collapses rather than individual pa rticles. This group of partic les is a cluster of a certain size and the powder deforms in clusters. An understanding of this clustering is very important to understand powder flow. It is fo r example a key factor in processes such as mixing. The stick-slip length is indicative of th e cluster size. Therefor e it is proposed that stick-slip can be used as an indicator for the length scal e of processes in a powder during flow and thus for the coherence length in a powder. The proposed stick-slip mechanism explains th e critical stick-slip velocity that is often observed by other research ers (Bucklin et al., 1996). Wh en a powder is deformed faster than the collapse velocity, there will be no drop in stress dur ing a collapse and the powder will deform continuously. Therefore stickslip is not exhibited above the critical velocity which is the collapse velocity. The powder will still deform in clusters though. Therefore it is propositioned th at all powders inherently have stick-slip but that not all powders exhibit it.

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133 The friction between the particles is an important property for the formation of stable powder structures. Atomic force mi croscope (AFM) measurements of the interparticle forces between polystyrene spheres re veal that the friction between the particles is non-linear, following a JKR contact model. The friction force betw een the particles is much larger than macroscopic friction e xperiments show. The results support the proposed stick-slip mechanism. The high friction, which includes adhesion forces, enables the polystyrene particle s to form stable structures in the powder and thus exhibit stick-slip. The inherent cohesion in a powder is due to van der Waals attractive forces between particles. The AFM is used to m easure the adhesion between particles. The measured adhesion force agrees with the th eoretical van der Waals force between polystyrene particles. Therefore, the van de r Waals force can be used to describe the particle adhesion between polystyrene particles. Discrete element simulations of a shear cell are performed, using the measured friction and theoretical van der Waals for ce. Although the simula tion was in 2D, the results match up well with the experimental r esults. The simulations show that the high inter-particle friction causes the powder to exhibit stick-slip, proving the proposed mechanism. The friction between the particles enables the form ation of a stable structure in the powder, which collapses during a slip event. In this research it is shown that a bo ttom up approach can be used to predict powder flow and more specifica lly stick-slip. An atomic fo rce microscope is used to measure particle properties. These properties are used as input parameters for discrete

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134 element simulations. With the right input para meters discrete element simulations can be used to predict powder flow and stick-slip. Suggestions for Future Work The majority of this work is based on one material, polystyrene powder. The reason for this is that the powder is very spherica l, enabling discrete element simulations. In powder technology it is dangerous to build th eories on the results of a single powder since all powders behave di fferently. When starting from first principles though, all powders should behave according to the rules. The challenge is to find out what physical or chemical processes are dominant for a ce rtain powder. Different powders should be investigated to find the b asic equations of phenomena such as non-elastic particle contacts, electrostatic ch arging of particles, non-s pherical particles, etc. Discrete Element Method simulations that have been used in this work are somewhat simplified. To start, the simulation is two dimensional, which is maybe reliable qualitatively, but not quantita tively. The contact model is elastic, disregarding plastic deformations between particles. For th e polystyrene powder this might be a good approximation, but for other powders it is not The particles in the simulation are all spherical, which is usually not the case in real ity. To make realistic simulations that apply to a wide range of powders, a program s hould be developed that can handle 3D simulations of non-spherical particles, with diffe rent contact models. A good particle interaction model needs to be established to implement in the DEM code. The model needs to correctly take hum idity, electrical forces, temperature, and other environmental and material characteristics in to account. The AFM is a useful tool for this and much progress is expected in the next years.

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135 The focus should not be that faster computers make be tter simulations. If the established inter-particle models are very complex, costing extensive computer time, it might be possible to take a shortcut. Not all particles in a process have to be simulated. The process can be divided in different system s and for every system a correct size has to be simulated. For example to effectively predic t the effect of moistu re on flow in a silo the following sequence could be used. Mois ture adsorption and mi gration on particle surfaces can be simulated on part of a particle surface, using the material properties of the particle and the liquid. This will give info rmation to simulate liquid bridge formation between two particles as a function of air humidity. Th e extra strength this bridge provides can be included in the particle in teraction model. Using DEM simulations, these interaction models will provide realistic r esults for small quantities of powder. These powder results can be fed into FEM simulati ons to simulate the flow through the silo. The power of this system is the capability to predict powder flow behavior, based on the properties measured with an atomic fo rce microscope using only a few particles. This enables the modification or design of pa rticles at an early st age of development, reducing costs down the line of the devel opment of a powder or process.

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136 APPENDIX A CALIBRATION SCHULZE TESTER Since the Schulze load cells are used both in compression and tension, the compliance in both directions was determined Figure A-1 shows that the compliance is different for tension and comp ression. Furthermore, the load cells show hysteresis for compression. This turned out to be no problem since the load cells were always unloaded before a measurement, so the loading cu rve was used for the calibration of the compression experiments. 0 5 10 15 20 024681012Reading (cm)Force (N) tension compression Figure A-1: Force calibration graph for the Schulze cell in compression and tension.

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137 APPENDIX B PARTICLE SIZE DISTRI BUTION POLYSTYRENE The particle size distribution of the 9 micron and 40 micron polystyrene powder are measured with the Coulter LS13320. The 9 micron powder shows a bimodal distribution. The mean of this powder is actu ally 7.7 micron, but the mean of the larger peak is 9 micron. Most of the work in th is dissertation is performed with the 40 micron powder which has a narrow size distribution. 0 5 10 15 20 25 0.010.1110100 Particle Size (Micron)Differential Volume (%) 40 m 9 m Mean: 40.49 7.71 S.D.: 6.18 4.15 d10: 32.95 0.34 d50: 40.58 7.83 d90: 49.72 12.73 40 micron polystyrene powder 9 micron polystyrene powder Figure B-1: Particle size distribution of polystyrene powders.

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138 APPENDIX C STIFFNESS CALCULATION SCHULZE TESTER To verify that the stick-slip signal is a powder signal and not a tester artifact, the stiffness of the system has been estimated. Th e top lid of the tester and the crossbar are very sturdy their stiffness is much higher than that of the load cells and the tension bars. The calculation is performed for an experi ment where the stick-slip signal had a magnitude of 40 N, converting to 20 N per te nsion bar. The displacement during the jump was 100 m. The stretch dL in a section of the tension bars with length L can be calculated using equation C-1, where F is the force, A the cross-sectional area, and E Young’s modulus of elasticity. The tension ba rs are modeled as shown in Figure C-1. L AE F dL C-1 397 mm 6.7 mm 8 m m 14.9 mm thickness 2.0 mm Figure C-1: Drawing of a te nsion bar (top) and schematic representation for stiffness calculation (bottom). Young’s modulus of elasticity fo r aluminum is taken as 70*109 N/m2. The elastic stretch due to the 20 N force change will be 3.8 m for the wide part of the bar and 0.17 m for the narrow part of the bar, so th e total stretch in a tension bar is 4.0 m.

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139 The deflection of the load cells is rate d at 0.29 mm at 20 kg, which corresponds to 24 m at a load of 20 N. The total deflection of the load cell plus tension bar is 28 m. This is four times smaller than the measured deflection of 100 m. The calculations show that the stick-slip si gnal can not be solely attributed to the stiffness of the tester system. About 75% of the elasticity of the tester with powder system comes from the powder.

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140 LIST OF REFERENCES Adams, M. J., Briscoe, B. J., Pope, L. (1987). A contact mechanics approach to the prediction of the wall friction of powders. in: Briscoe, B. J. & Adams, M. J. (Eds.), Tribology in particulate technology. A. Hilger, Bristol, England. pp. 8-22. Airey, D. W., Wood, D. M. (1988). The Cambridge true tria xial tester. in: Donaghe, R. T., Chaney, R. C. & Silver, M. L. (Eds.) Advanced triaxial testing of soil and rock. ASTM, Philadelpia, USA. pp. 796-805. Akers, R. J. (1992). The certi fication of a limestone for jeni ke shear testing. Community Bureau of Reference, Brussels, Belgium. Albert, I., Tegzes, P., Albert, R., Sample, J. G., Barabasi, A. L., Vicsek, T., Kahng, B., Schiffer, P. (2001). Stick-slip fluctuations in granular drag. Physical Review E, 64(3), art. no.-031307. Arthur, J. R. F., Menzies, B. K. (1972). I nherent anisotropy in a sand. Geotechnique, 22(1), 115-128. ASTM (2000). Standard test method for shear testing of bulk solids using the Jenike shear cell. in: Annual bo ok of ASTM standards, 04.09 no D6128-00. ASTM, Philadelphia, USA. ASTM (2002). Standard shear test method for bulk solids using the Schulze ring shear tester. in: Annual book of ASTM standards, 04.09 no D6773-02. ASTM, Philadelphia, USA. Berman, A. D., Ducker, W. A., Israelachvili, J. N. (1996). Origin a nd characterization of different stick-slip friction mech anisms. Langmuir, 12(19), 4559-4563. Bhushan, B. Ed. (1999). Handbook of micro/na notribology. The mech anics and materials science series. CRC Press, Boca Raton, USA. Bhushan, B. (2002). Introduction to tribol ogy. John Wiley & Sons, New York, USA. Bilgili, E., Yepes, J., Stephenson, L., Johanson, K., Scarle tt, B. (2004). Stress inhomogeneity in powder specimens tested in the Jenike shear cell: Myth or fact? Particle & Particle Systems Ch aracterization, 21(4), 293-302. Binnig, G. (1992). Force microsc opy. Ultramicroscopy, 42, 7-15.

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141 Binnig, G., Gerber, C., Stoll, E., Albrecht, T. R., Quate, C. F. (1987). Atomic resolution with atomic force microscope. Eu rophysics Letters, 3(12), 1281-1286. Binnig, G., Quate, C. F., Gerber, C. (1986). Atomic force microscope. Physical Review Letters, 56(9), 930-933. Binnig, G., Rohrer, H. (1983). Scanning tunne ling microscopy. Helvetica Physica Acta, 56(1-3), 481-482. Binnig, G., Rohrer, H., Gerber, C., Weib el, E. (1982). Surface studies by scanning tunneling microscopy. Physical Review Letters, 49(1), 57-61. Bogdanovic, G., Meurk, A., Rutland, M. W. (2000). Tip friction torsional spring constant determination. Colloids and Surfaces B-Biointerfaces, 19(4), 397-405. Bowden, F. P., Tabor, D. (1950) The friction and lubrication of solids. Clarendon Press, Oxford, UK. Bucklin, R. A., Molenda, M., Bridges, T. C., Ross, I. J. (1996). Slip-stick frictional behavior of wheat on galvan ized steel. Transactions of the ASAE, 39(2), 649-653. Budny, T. J. (1979). Stick-slip friction as a method of powder flow characterization. Powder Technology, 23(2), 197-201. Burnham, N. A., Kulik, A. J. (1999). Surface fo rces and adhesion. in: Bhushan, B. (Ed.), Handbook of micro/nanotrib ology. CRC Press, Boca Rat on, USA. pp. 247-271. Cain, R. G., Page, N. W., Biggs, S. (2001) Microscopic and macroscopic aspects of stick-slip motion in granular shear. P hysical Review E, 64(1), art. no.-016413. Coulomb, C. (1773). Application des regles de maximis a quelques problems de statique relatifs a l'archite cture. Memoires desavants etrange rs de l'Academie des Sciences de Paris. Cundall, P. A., Strack, O. D. L. (19 79). Discrete numerical-model for granular assemblies. Geotechnique, 29(1), 47-65. Derjaguin, B. V., Muller, V. M., Toporov, Y. P. (1975). Effect of contact deformations on adhesion of particles. Journal of Collo id and Interface Science, 53(2), 314-326. Drucker, D. C., Gibson, R. E., Henkel, D. J. (1957). Soil mechanics and plastic analysis or limit design. Transactions of the ASCE, 122, 338-346. Drucker, D. C., Prager, W. (1952). Soil mech anics and plastic analysis or limit design. Quarterly of Applied Mathematics, 10(2), 157-165.

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142 EFCE (1989). Standard shear testing technique for partic ulate solids using the Jenike shear cell. Working Party on the Mechanics of Particulate Solids. The Institution of Chemical Engineers, Rugby, UK. Ennis, B. J., Green, J., Davies, R. (1994). Th e legacy of neglect in the United-States. Chemical Engineering Progress, 90(4), 32-43. Esayanur, M. S. S. (2005). Interparticle force based methodology for prediction of cohesive powder flow properties. Ph .D. dissertation. Material Science and Engineering, University of Florida, Gainesville, USA. Feda, J. (1982). Mechanics of particulate mate rials : The principles. Elsevier Scientific, Amsterdam, The Netherlands. Hunston, M. (2002). Innovative thin-film pressure mapping sensors. Sensor Review, 22(4), 319. Hvorslev, M. J. (1937). ber die festigkeitseigenschafte n gestrter bindiger bden. Ingeniorvidenskab, Skrifter A 45. Israelachvili, J. N. (1985). Intermolecular and surface forces. Academic Press, London, UK. Itasca Manual (2002). Verification problems and example applications, Itasca Consulting Group, Minneapolis, USA. Janssen, H. Z. (1895). Versuche ber gestreid erdruck in silozellen. Zeitschrift des VDI, 39, 1045-1049. Janssen, R. J. M. (2001). Structure and shear in a cohesive powder. Ph.D. dissertation. Department of Chemical Engineering, De lft University of Technology, Delft, The Netherlands. Janssen, R. J. M., Verwijs, M. J., Scarlett, B. (2005). Measuring flow functions with the flexible wall biaxial tester. Po wder Technology, 158(1-3), 34-44. Jenike, A. W. (1961). Gravity flow of bulk so lids. Bulletin 108. University of Utah, Salt Lake City, USA. Jenike, A. W. (1964). Storage a nd flow of solids. Bulletin 1 23. University of Utah, Salt Lake City, USA. Johanson, K., Bucklin, R. (2004). Measure ment of k-values in diamondback hoppers using pressure sensiti ve pads. Powder Tec hnology, 140(1-2), 122-130.

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143 Johnson, K. L., Kendall, K., Roberts, A. D. (1971). Surface energy and contact of elastic solids. Proceedings of the Royal Soci ety of London Series a-Mathematical and Physical Sciences, 324(1558), 301-313. Kanatani, K. I. (1984). Distribut ion of directional-da ta and fabric tens ors. International Journal of Engineering Science, 22(2), 149-164. Kolb, E., Mazozi, T., Clement, E., Duran, J. (1999). Force fluctuati ons in a vertically pushed granular column. European Physical Journal B, 8(3), 483-491. Lacombe, F., Zapperi, S., Herrmann, H. J. (2000). Dilatancy and friction in sheared granular media. European Phys ical Journal E, 2(2), 181-189. Leuenberger, H., Lanz, M. (2005). Phar maceutical powder technology-from art to science: The challenge of the FDA's pr ocess analytical technology initiative. Advanced Powder Technology, 16(1), 3-25. Li, F., Puri, V. M. (1996). Measurement of anisotropic behavior of dry cohesive and cohesionless powders using a cubical triaxial tester Powder Technology, 89(3), 197-207. Merrow, E. W. (1985). Linking Rand-D to problems experien ced in solids processing. Chemical Engineering Progress, 81(5), 14-22. Merrow, E. W. (1988). Estimating startup times for solids-processing plants. Chemical Engineering, 95(15), 89-92. Merrow, E. W. (2000). Problems and progr ess in particle processing. Chemical Innovation, 30(1), 34-41. Meurk, A. (2000). Force measurements using scanning probe microscopy : Applications to advanced powder processing. Ph.D. dissertation. Institutionen fr kemi, Tekniska hgskolan i Stockholm, Stockholm, Sweden. Molerus, O. (1978). Effect of interparticle cohesive forces on flow behavior of powders. Powder Technology, 20(2), 161-175. Molerus, O. (1996). History of civilisation in the western he misphere from the point of view of particulate technology. Adva nced Powder Technology, 7(1), 71-77. Oda, M., Konishi, J., Nematnasser, S. (1980) Some experimentally based fundamental results on the mechanical-behavior of granular-materials. Geotechnique, 30(4), 479-495.

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144 Rabinovich, Y. I., Adler, J. J ., Ata, A., Singh, R. K., Moudgi l, B. M. (2000a). Adhesion between nanoscale rough surf aces I. Role of asperity geometry. Journal of Colloid and Interface Science, 232(1), 10-16. Rabinovich, Y. I., Adler, J. J., Ata, A., Singh, R. K., Moudgil, B. M. (2000b). Adhesion between nanoscale rough surf aces II. Measurement and comparison with theory. Journal of Colloid and Inte rface Science, 232(1), 17-24. Reynolds, O. (1885). On the dilatancy of medi a composed of rigid particles in contact. Philosophical Magazine, 20, 127. Robbins, M. O., Thompson, P. A. (1991). Critical velocity of stick-slip motion. Science, 253(5022), 916-916. Roberts, A. W., Wensrich, C. M. (2002). Flow dynamics or 'quaking' in gravity discharge from silos. Chemical Engin eering Science, 57(2), 295-305. Roberts, I. (1882). Determination of the ve rtical and lateral pr essures of granular substances. Proceedings of the R oyal Society of London, 36, 225-240. Roscoe, K. H., Schofield, A. N., Wroth, C. P. (1958). On the yielding of soils. Geotechnique, 8, 22-53. Rothenburg, L., Bathurst, R. J. (1989). An alytical study of induced anisotropy in idealized granular-materials Geotechnique, 39(4), 601-614. Saraber, F., Enstad, G. G., Haaker, G. ( 1991). Investigations on the anisotropic yield behavior of a cohesive bulk solid Powder Technology, 64(3), 183-190. Scarlett, B. (2002). Particle technology th e 4M business. D.Sc. dissertation. Department of Chemical Engineering, L oughborough University of Technology, Loughborough, UK. Scarlett, B., van der Kraan, M., Janssen, R. J. M. (1998). Porosity: A parameter with no direction. Philosophical Tr ansactions of the Royal Society of London Series AMathematical Physical and Engi neering Sciences, 356(1747), 2623-2648. Schulze, D. (2003). Timeand velocity-depende nt properties of powders effecting slipstick oscillations. Chemical Engineer ing & Technology, 26(10), 1047-1051. Schwedes, J. (2003). Review on testers fo r measuring flow properties of bulk solids (based on an IFPRI-report 1999). Gr anular Matter, 5(1), 1-43. Thompson, P. A., Robbins, M. O. (1990). Or igin of stick-slip motion in boundary lubrication. Scien ce, 250(4982), 792-794.

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145 van der Kraan, M. (1996). Techniques for th e measurement of cohesive powders. Ph.D. dissertation. Department of Chemical Engineering, Delft University of Technology, Delft. Verwijs, M. J. (2001). A macr oscopic view on powder flow behavior. Master thesis. Department of Chemical Engineering, De lft University of Technology, Delft, The Netherlands. Verwijs, M. J., Abdel-Hadi, A. I., Cristescu, N. D., Scarlett, B. (2003). Comparison of a cylindrical and cubical biaxial powder tester. Proceedings 4th Conference for Conveying and Handling of Particulate Ma terial, Budapest, Hungary, 1, pp. 5.675.72. Verwijs, M. J., Janssen, R. J. M., Scarlett, B. (2002). Influence of the intermediate principal stress during powder shear in a biaxial tester. Proceedings Fourth World Congress on Powder Technology, Sydney, Australia, paper 135. Volfson, D., Tsimring, L. S., Aranson, I. S. (2004). Stick-slip dynamics of a granular layer under shear. Physical Re view E, 69(3), art. no.-031302. Wong, R. K. S., Arthur, J. R. F. (1985). Induced and i nherent anisotropy in sand. Geotechnique, 35(4), 471-481. Yoshizawa, H., Israelachvili, J. (1993). Funda mental mechanisms of interfacial friction. 2. Stick-slip friction of spherical and chain molecules. Journal of Physical Chemistry, 97(43), 11300-11313. Zhupanska, O. I., Verwijs, M. J., Scarlett, B. (2003). Anisotropy in powders: From microto macroscale. Proceedings annua l AIChE meeting, Sa n Francisco, USA.

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146 BIOGRAPHICAL SKETCH Marinus Jacobus (Marco) Ve rwijs was born in Moerkapelle, The Netherlands, on March 19, 1976. He finished his high sc hool education from C.S.G. Damstede in Amsterdam. In August 1994 Marco was admitte d into the chemical engineering program at the Delft University of Technology. He earned a Master of Science degree in August 2001 with the thesis A Macroscopic view on Powder Flow Behavior. In October 2001, he came to the United States of America to conduct research at the Engineering Research Center for Particle Science and Technology at the University of Flor ida. In January 2002 he joined the Department of Ch emical Engineering at the Univ ersity of Florida in pursuit of a doctoral degree. Marco expects to obtai n a Doctor of Philosophy degree in chemical engineering in the fall of 2005. After graduation he plans to join Transform Pharmaceuticals, Inc., a Johnson & Johnson comp any, in Lexington, MA, as a scientist in the product development group.


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Title: Stick-Slip in Powder Flow: A Quest for Coherence Length
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Copyright Date: 2008

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Title: Stick-Slip in Powder Flow: A Quest for Coherence Length
Physical Description: Mixed Material
Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
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STICK-SLIP IN POWDER FLOW: A QUEST FOR COHERENCE LENGTH


By

MARINUS JACOBUS VERWIJS













A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005





























Copyright 2005

by

Marinus Jacobus Verwij s

































This document is dedicated to professor Brian Scarlett.
















ACKNOWLEDGMENTS

I would like to acknowledge my advisors, Dr. Kerry Johanson and Dr. Spyros

Svoronos, for their guidance and support. Dr. Johanson's extensive experience and

knowledge in powder technology were invaluable sources of information. Dr. Svoronos'

outside view and critical approach were very useful and his help in administrative

manners was indispensable. My special gratitude goes to professor Brian Scarlett who

always showed his trust in me. I tnank him for convincing me to pursuit a doctoral study

and to encourage me to make my own path.

Thanks are also due to all the other faculty members who helped me with

discussions and suggestions. I thank my committee members Dr. Anuj Chauhan, Dr.

Jennifer Curtis, and Dr. Ray Bucklin for their helpful suggestions. I enjoyed the

collaboration with Dr. Bucklin in the development of a new, large size, Jenike Tester.

I would like to thank Dr. Brij Moudgil and the National Science Foundation's

Engineering Research Center for Particle Science and Technology and our industrial

partners for their financial support. I would also like to thank the present and past

administration and staff of the Particle Engineering Research Center for their help during

my stay.

I would like to gratefully acknowledge my group members including Dr. Nicolaie

Cristescu, Dr. Yakov Rabinovich, Dr. Olesya Zhupanska, Dr. Claudia Genovese, Dr. Ali

Abdel-Hadi, Dr. Ecivit Bilgili, Dr. Mario Hubert, Dr. Dimitri Eskin, Dr. Nishanth

Gopinathan, Dr. Madhavan Esayanur, Dr. Caner Yurteri, Dauntel Specht, Rhye Hamey,









Maria Palazuelos Jorganes, Stephen Tedeschi, Milorad Djomlija, Osama Saada, Julio

Castro, Benjamin James, Mark Pepple, and Bill Ketterhagen for their help in carrying out

my research, and also for their support and encouragement. I would like to thank Scott

Brown for his help in several matters. Many thanks go to the undergraduate students who

helped me with my research, especially Aaron Gfeller and Jesse Schrader.

I thank my parents for always believing in me and trusting me. They have always

been there for me when I needed them. I apologize that I do not pursuit my endeavors

closer to their house so I can see them more often.

My time in Gainesville would not have been as enjoyable without all my friends at

the PERC, Team Florida, and the Trigators and I thank them for that. I thank Rhye for

being a great roommate for four years. Lastly but mostly I want to thank my girlfriend

Dauntel for being so patient with me.
















TABLE OF CONTENTS



ACKNOW LEDGM ENTS ........................................ iv

LIST OF FIGURES ..................................... ix

ABSTRACT................................. .............. xiv

CHAPTER

1 INTRODUCTION ................... .................. .............. .... ......... .......

Solids H handling Problem s in Industry ........................................................................2
ScientiSic Approach to Powder Technology.........................................................4
Stick-slip in Pow der Flow ................................................. ...............6
Focus of Dissertation .................. ........... .. ........ ................ ..... .7
Outline of Dissertation.................................8

2 PO W D ER TE CH N O L O G Y .................................................................................. 10

Powder Flow...................................... .................. ............. .........10
M acroscopic Scale ............... ............................................................ ..... ................. 12
Pow der M echanics. .................. .... .................................. .... .. ............ 12
Constitutive M odels............... ............................. ........... .... .. ... ...... 16
Pow der Structure ................ ........ ............................. ...... .... .. .. ...... 19
Powder Testers .......................... ......................20
Direct Shear Cell M easurements ...................... ......................................... 22
Microscopic Scale................... .......................... 24
Pressure M apping ............................. ....................24
Inter-Particle Forces ............................................. .... .....25
Contact M echanics .............................................. ..... .. 29
D iscrete Elem ent M odeling........................................................................... ...... 33
Fabric Tensors ........................... ............... .........34

3 INHOMOGENEITY AND ANISOTROPY IN SHEAR TESTERS ......................37

E effective Y field L ocus .................. .................. ........................ .................. 37
Flow Function ...................................... ....... ........... .43
In flu en ce of M oistu re ..........................................................................................4 3
D irectional D ependence .............................................. ............... 44


ield L ocu s ........................................................................ .................... 37
F lo w F u n ctio n ........................................................................................................ 4 3
Influence of M oisture ......................................... ............................ ...............43
D irectional D ependence ........................................ .........................................44









A ngle of Internal Friction ...................... ..... ............................ ............... 46
N orm al Stress D distribution ........................................................ ...... ......... 48
G eom etric A nisotropy .................................................. ........................... 49
Modeling of Anisotropy in Biaxial Shear................................................. 52
B iaxial E xperim ents ................................................. ............................... 52
Set-up B iaxial Sim ulation......................................................... ............... 54
Stress and Porosity Distributions..................... .............................. 55
M ic ro stru ctu re ............................................................................................... 5 8

4 UNIAXIAL COMPACTION OF POWDERS ................................. ...............61

D esign of the U niaxial T ester......................................................................... ...... 62
U niaxial Experim ents ................................... ........................................ 65
Uniaxial Compaction of Microcrystalline Cellulose..................................66
Uniaxial Compaction of Polystyrene Powder .............................................67
M odeling of U niaxial Com action .................................. .............................. ....... 69
Influence of C ell G eom etry ......................................... ....................... ........... 70
Pseudo-Stress Control versus Strain Control ............................................... 74

5 STICK-SLIP IN POWDER FLOW................................................................. 76

Introdu action to Stick -Slip ........................................ .............................................77
E xperim mental Setup ............. ......................................................................... .. ... 82
R results of Schulze Tests ............................................... .... .... .. ........ .... 83
Influence of N orm al Stress ........................................................ ............... 87
Influence of Shear V elocity.......................................................... ............... 89
Influence of Particle Size............. ................................... ..................91
Influence of Moisture Content ...............................................93
R results of U niaxial T ests ................................................................. .............. 96
M mechanism of Stick-slip ............................................................................98

6 INTER-PARTICLE FORCE MEASUREMENTS .............................................101

A tom ic F orce M icroscope ............................................................. ..................... 102
E x p erim ental S etu p ............................................. ........................................ 10 9
A dhesion Force M easurem ent.......................................... ......... ... ............... 110
Friction Force Measurement... .. ................... ....................111
Inter-particle Force Modeling ........................... ................... 116
Atomic Force Measurement Simulation.....................................................117
Friction M modeling .................. ........................... .... .. .. .. ........ .... 119
Stick-slip M odeling ..................................... .... ........ .. .. .. .. .. ........ .... 123

7 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK..........................129

C o n clu sio n s......................................................... ................ 13 0
Suggestions for Future W ork ........................................................................ ... ... 134









APPENDIX

A CALIBRATION SCHULZE TESTER ............................. ......... .............136

B PARTICLE SIZE DISTRIBUTION POLYSTYRENE .......................................137

C STIFFNESS CALCULATION SCHULZE TESTER................. ...................138

L IST O F R E F E R E N C E S ...................................................................... ..................... 140

BIOGRAPHICAL SKETCH ............................................................. ...............146















LIST OF FIGURES


Figure page

1-1 Comparison of planned and actual startup time for solids handling plants. ............3

1-2 Average startup time (left) and performance (right) of plants ................................4

1-3 Pyram id of know ledge. ........................................................................... .............5

2-1 Hvorslev diagram .. ....................... ......... .. .. ..... .......... .... 11

2-2 Fam ily of yield loci ....................... .. ....... .................. ............ 12

2-3 Stresses on a powder sample with the corresponding stress tensor .......................14

2-4 Schematic of deformation with corresponding strain tensor............... .......... 16

2-5 Possible shapes of the failure surface; hexagonal (left) and triangular (right).........18

2-6 Operating window for stress control of a biaxial (right) and von Karman tester.... 21

2-7 Schem atic of the Schulze tester ..................................................................... .... 22

2-9 Schem atic of a Tekscan sensor pad ..................................... ......... ..... .......... 25

2-10 Force-distance relationships for the Born repulsion, London van der Waals
attraction, electrostatic repulsion, and combinations of those...............................28

2-11 Schematic representation of a liquid bridge between a sphere and a rough plate....29

2-12 Schematic representation of Hertzian and JKR contact radius and deformation.....31

2-13 Angular distribution of parameter K, near isotropic (left) and anisotropic (right). .35

3-1 Critical Mohr circles with tangent effective yield locus for BCR Limestone,
M measured w ith a Schulze Cell ............................................................................ 38

3-2 Effective yield loci of 40 [m polystyrene powder for different powder moisture
contents, m measured with a Schulze cell ........................................ ............... 39

3-3 Schematic representation of the critical strength (fc*) of a powder .....................40









3-4 Method of computing critical strength from flow function...............................41

3-5 Effective angle of internal friction according to Jenike. .......................................42

3-6 Flow function of polystyrene powder for different moisture contents.....................44

3-7 Comparison of strength measurements of BCR limestone with the Schulze cell
(PERC) with Akers and Saraber et al. ......................................... 45

3-8 Flow function of BCR Limestone for a standard and reverse experiment...............46

3-9 The angle of internal friction (q) as a function of the major principal stress for a
standard and reverse experiment. .. .............. ................. ............47

3-10 Typical normal stress profile at the bottom of a slice of the Schulze cell.............48

3-11 Normal force on the bottom of a Schulze cell during steady state deformation of
BCR limestone. ..................................................49

3-13 Stress profile at the bottom of a Jenike cell during pre-shear and failure in the
forw ard direction, using silica pow der. .............................................. .....51

3-13 Stress profile at the bottom of a Jenike cell during pre-shear in the forward
direction and failure in the backward direction.................................................51

3-14 Typical result of a standard experiment with the Flexible Wall Biaxial Tester.......53

3-15 Typical result of an anisotropic experiment with the Flexible Wall Biaxial
Tester...............................................................53

3-16 The direction of wall movement at different stages of the biaxial test simulation. .55

3-17 Normal stresses and porosity distributions after biaxial consolidation................55

3-18 Normal stresses and porosity distributions after pre-shear. ..................................56

3-19 Normal stresses and porosity distributions after forward failure..........................58

3-20 Normal stresses and porosity distributions after anisotropic failure. .......................58

3-21 Normal contact force intensity distribution after different steps of the biaxial
simulation. ........................... ............................59

4-1 Picture of the Uniaxial Tester................................ ...............62

4-2 Top view picture of the Uniaxial Tester.................... .......................... 63

4-3 Different sample holder box designs....................... .................... 64


al Tester............ ......... ....... ............... 63

4-3 Different sample holder box designs........................... ............. ............. 64









4-4 Uniaxial compaction of Microcrystalline Cellulose PH101. ...............................67

4-5 Stress controlled uniaxial compaction and relaxation curve of 40 |tm polystyrene
p o w d e r ........................................................................... 6 8

4-6 Strain controlled uniaxial compaction curve of 40 |tm polystyrene powder. .........68

4.6 Sample box geometries used for the simulations. .................................................70

4-7 Overall stress on the x- and y-walls of two cell geometries during uniaxial
com action. .......................................... ............................ 71

4-8 Distributions of normal stress in the x-direction for geometry 1 (left) and
geom etry 2 (right) ..................................... ............................ .. ....... 72

4-9 Fabric tensors for uniaxial compaction using two different geometries ................73

4-10 Consolidation curves for uniaxial simulations. .................. ....................... 75

5-1 Schematic representation of stick-slip due to surface roughness...........................78

5-2 Schematic representation of the bifurcation of the friction force as a function of
th e sh ear v elo city .................................................................. ............... 7 9

5-3 Different types of stick-slip ...................................... ......... .... ............... 81

5-4 SEM images of cornstarch (left) and polystyrene particles (right). .........................82

5-5 Typical stick-slip signal from a Schulze cell. .................................. .................84

5-6 Picture of the potentiometer fixture to the Schulze cell lid................................84

5-7 Typical signals for the horizontal top lid displacement (grey, left axis) and shear
stress (black, right axis) during stick slip in a Schulze cell...................................86

5-8 Typical horizontal displacement of the top lid relative to the base during stick-
slip in a Schulze cell ..................................................... .. ....................... 86

5-9 Maxima, minima, and magnitude of stick-slip as a function of normal stress for
the 40 |tm polystyrene pow der. ........................................ ........................... 87

5-10 Power plot from a Fast Fourier Transform of a low stress experiment (left) and a
high stress experim ent (right) ...................................................................... ...... 88

5-11 Frequency of stick-slipfas a function of the applied normal stress o, for 9 |tm
polystyrene pow der. ..................... .................. ................. .... ....... 89

5-12 Influence of shear velocity on the stick-slip frequency of 9 |tm polystyrene
powder m measured in the Schulze cell. ........................................... ............... 90









5-13 Stick-slip length scale of 9 |tm polystyrene powder as a function of normal
stress for three different shear velocities of the Schulze cell................................91

5-14 Characteristic length scale of stick-slip in number of particle diameters as a
function of the normal stress for the 9 and 40 [pm polystyrene powder .................92

5-15 Magnitude of stick-slip as a function of the normal load for different moisture
contents in 40 [pm polystyrene powder with linear trend lines..............................94

5-16 Slope of the trend lines in Figure 5-15 as a function of the moisture content..........95

5-17 Characteristic stick-slip length as a function of the normal stress for several
m oisture contents........... .............................................................. ......... ........ 96

5-18 Detail of Figure 4-6 at a low stress level (left) and a high stress level (right). .......97

5-19 Standard deviation of stick-slip signal as a function of the axial stress .................97

5-20 Frequency of stick-slip signal as a function of the stress and strain ......................98

6-1 Schematic of an Atomic Force Microscope. .................................. ...............102

6-2 Schematic of lateral force measurement with the AFM showing torsional
deflection or twisting of a rectangular cantilever ..................................................103

6-3 Typical AFM result for force interaction measurements between a tip and a
substrate..................................... ................................. .......... 106

6-4 Schematic showing the relative separation between a cantilever tip and the
substrate..................................... ................................. .......... 107

6-5 Schematic of the cantilever, tip, and substrate, represented by springs and
dashpots.............. ............. .. ................ ............. ........... 108

6-6 Pictures of particles attached to a cantilever (left and middle, 500x
magnification) and the substrate of particles (right, 200x magnification). ............109

6-7 Typical force-displacement result for polystyrene particles. ...............................110

6-8 Typical AFM friction loop of a measurement between two polystyrene particles. 112

6-9 Friction force as a function of the external normal force for polystyrene
particles. .................................... ........................... ...... .......... 113

6-10 Friction force as a function of the total normal force for polystyrene particles.....115

6-11 Discrete element model of the AFM cantilever with attached polystyrene
particle and a polystyrene substrate particle. ....................................................... 118









6-12 Discrete element simulation of a pull-off AFM experiment using the DMT
m odel .................... ..... .... .......... ...........................................119

6-13 Friction loop measured with a DEM simulation, simulating polystyrene particles
w ith a H ertzian contact m odel ................................................................... ....... 120

6-14 Vertical cantilever deflection during a DEM simulation of a friction loop..........122

6-15 Geometry of the Jenike cell in the DEM simulation................................124

6-16 Shear force as a function of displacement during a Jenike cell simulation ..........125

6-17 Comparison of shear experiment simulations. ............................... ......... ...... 126

6-18 Shear force as a function of the displacement with a friction coefficient of 0.3.... 127

A-1 Force calibration graph for the Schulze cell in compression and tension..............136

B-l Particle size distribution of polystyrene powders. ............................................... 137

C-1 Drawing of a tension bar (top) and schematic representation for stiffness
calculation (bottom ). ........................ ....... ..... .. ...... .............. 138















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

STICK-SLIP IN POWDER FLOW: A QUEST FOR COHERENCE LENGTH

By

Marinus Jacobus Verwij s

December 2005

Chair: Kerry Johanson
Cochair: Spyros Svoronos
Major Department: Chemical Engineering

Many industries such as the food, chemical, agricultural, and pharmaceutical

industries use powdered material, either as a product, feedstock, or intermediate.

Compared with liquids and gasses, processes involving powders encompass many

problems, causing reduced production and unscheduled downtimes. Most problems occur

in powder transfer processes. The problems stem from a lack of understanding of the

basic principles of powders.

Powder flow, which is the deformation of a bulk powder, is one method of powder

transfer which is not well understood. One specific problem is stick-slip, which is a

discontinuous or stepwise flow of powder. Stick-slip is a known phenomenon from the

field of tribology, but there is no understanding of stick-slip in powders. In industry,

stick-slip can cause structures to vibrate violently, causing noise and structural damage. It

also causes problems in mixing, creating serious problems in the product quality.










This research investigates stick-slip and proposes a mechanism that causes stick-

slip in powders. The influence of particle size, moisture content, and stress is

investigated. The experiments are conducted with a Schulze shear tester and a newly

developed uniaxial tester. The performance of those testers is first established by

measurements and simulations.

The shear stress during stick-slip is recorded and the magnitude and frequency of

the signal are calculated. Discrete element modeling is used to simulate powder flow

processes that are investigated. The results show that powder cohesion and inter-particle

friction are key factors for stick-slip.

Stick-slip events prove to have a characteristic length scale, which can be

correlated with the particle size and cohesion. It is proposed that a slip event is a collapse

of the structure of the powder. The structure of the powder is built up from clusters of

particles. The size of these clusters defines the characteristic stick-slip length. The cluster

size is a function of the cohesion of the powder. The characteristic length is called the

coherence length. The coherence length is a measure of the sphere of influence of

individual particles into the powder bulk. It is a parameter that can be used to distinguish

between the discontinuous and continuous scale of a powder.














CHAPTER 1
INTRODUCTION

Powder technology is one of the oldest professions in the world. It has been

developed in the fields of food preparation, ceramics, paint production, and construction

(Molerus, 1996). Some scientists believe that the great pyramids in Egypt were not built

by rolling enormous building blocks from quarries to the construction site, but by

forming the building blocks on-site. For this, granular limestone was poured into a

rectangular mold and hardened much like concrete.

It is remarkable that such an old field as powder technology is not developed up to

standard. Compared with gas and liquid handling, powder technology is far behind and

often still an art rather than a science. Part of the problem lies in the communication

between scientists and plant operators. Merrow (1985) linked research and development

to problems in industry and found that there is an information breakdown between

operation and basic research and theory building. Most of the problems in industry are of

a mechanical or physical type rather than a chemical. Therefore, scientists in the field,

who are mostly chemical engineers, believe that the problems need to be solved by the

operators. So research money from government or industry is not focused towards

solving particle problems fundamentally. Problems are often solved with patchwork

without trying to find a fundamental or permanent solution.

The stigma that powder technology does not belong to chemical engineering can be

found back in the curricula of engineering schools. First, not many schools teach particle

or powder technology to engineering students. When they do offer such courses, it is









often taught in the civil engineering or mechanical engineering department. Few

departments of chemical engineering prepare their students for the problems concerning

granular material which they will encounter during their professional career. Chemical

engineers are the ones who design the process plants and often operate them as well. In a

1994 publication it is stated that this neglect of particle technology in education is

especially serious in the USA (Ennis et al., 1994).

In his publication Merrow (1985) showed that the largest mechanical problem

causing plants to perform below design is failure of solids transfer. There are several

methods to transfer solids (i.e., particles), depending on the size and form of the solids:

slurries of solids in a liquid can be pumped, dry powders can be conveyed on belts or in a

pneumatic system, or can flow under gravity. This dissertation handles the flow of

powders. It specifically looks at discontinuous powder flow, which is called stick-slip.

Solids Handling Problems in Industry

The study by Merrow was performed in the 1980's and shows that plants handling

solids materials perform much worse than plants handling gasses and liquids (Merrow,

1985, 1988). About two thirds of the plants in the study operated at less than 80% of the

design and one third operated at less than 60% of the design. The average for the solids

handling plants was 64% of design capacity, compared to a 90 to 95% industry standard.

The plants also needed much longer startup times than anticipated. Figure 1-1 shows the

planned and actual startup times for plants with different feedstocks. Plants with a liquid

and/or gas feedstock producing solids needed a marginally longer startup time than

planned. A feedstock of refined solids, meaning the solids have undergone some type of

prior processing, show a startup time that is more than twice the planned time. When the

feedstock consists of raw materials, e.g., minerals, this worsens to an actual startup time










of 18 months, more than three times the planned time. Each month a startup slips behind

adds an average of $350,000 to the capital cost (in 1988). The actual cost of the delays

will be higher due to capital appreciation and market loss.

20
U Planned startup time
:F [OActual startup time
E16
0
E

E12
-t-
c8




0

Liquid-gas Refined-solids Raw-solids
Type of process plant

Figure 1-1: Comparison of planned and actual startup time for solids handling plants with
different types of feedstock (Merrow, 1988).

The study above is twenty years old and some of the data used in the study are 40

years old. After publication of the study more research money has been allocated towards

particle technology in the USA. A second study in 2000 showed that 15 years later there

had been some improvement (see Figure 1-2) (Merrow, 2000). The data in the Figure are

for plants starting up between 1996 and 1998. It is still apparent that plants handling

solids perform worse and need more startup time than plants handling liquids and gasses.

It should be noted that the 1985 study was for 40 plants in the USA and Canada while the

2000 study included 287 solids handling plants from around the world.

For solids handling plants basic technical data such as heat and mass balances are

often not available. Even if this information is available from other plants, it turns out

that it is difficult to transfer this process understanding from one plant to the other. There










is a lack of fundamental understanding of particle and powder behavior, which makes it

difficult to extrapolate knowledge from one facility to the other.

8 100
Q-

80-
06-
E
E0 0

14 2
(n S 40 -
F0)

S20
0
0 -- 0 -0-
1 2 3 4 1 2 3 4
Type of process plant Type of process plant

Figure 1-2: Average startup time (left) and performance (right) of plants handling only
solids and gasses (1), solid intermediates or products (2), refined solids feed
(3), and raw solids feeds (4) (Merrow, 2000).

Scientific Approach to Powder Technology

The key in solving problems in particle technology lies in understanding the basics.

When the fundamental behavior of powders is understood, reliable process and product

design is possible. In a recent paper, Leuenberger and Lanz (2005) review the state of the

art in the pharmaceutical powder technology. They state what has been known for some

time: Powder technology has to move from art to science. This is visualized in the

pyramid of knowledge (see Figure 1-3). Product and process design, as well as process

and quality control, have to move forward from a trial-and-error approach towards design

based on first principles. Therefore, research should be focused on the understanding of

first principles. For this it is often necessary to first understand the mechanistic behavior.

To work towards fundamental understanding Professor Scarlett initiated and

promoted the 4M business for particle technology (Scarlett, 2002). The four M's consist


k towards fundamental understanding Professor Scarlett initiated and

promoted the 4M business for particle technology (Scarlett, 2002). The four M's consist










of Making, Measuring, Modeling, and Manipulating. Measuring and modeling are tools

to design and control products and processes, which we want to make and manipulate. In

industry the focus is on making products or equipment to handle these products.

Manipulation of these products and processes can optimize them. The tools to do this are

measurements and models. As said, these models should be built upon first principles.

The input data for the models have to be measured with reliable testers. Therefore, the

development of models and measurements needs to go hand in hand; better

measurements make better models and better models make better measurements.


1st
Principle

MECHANISTIC
UNDERSTANDING

CAUSAL LINKS
PREDICT PERFORMANCE









Figure 1-3: Pyramid of knowledge (Leuenberger and Lanz, 2005).

To be able to use the 4M business properly, a systems approach needs to be used.

Every system has its own time and length scale. From bottom up we can for example

recognize atoms, molecules, particles, powders, process equipment, processes, and the

environment. Every system builds up the next, each with its distinct time and length

scale. To understand the first principles of a system, the first principles of the system it is

built from need to be understood.









In powder flow the knowledge level ranges from trial-and-error up to in some cases

a mechanistic understanding. The state of the art in the field is mainly based on the work

by Dr. Jenike (1961; 1964), which was developed in the 1960's. In his work, a

mechanistic model is used based on Mohr-Coulomb behavior of powders. The model

uses data from shear testers to predict powder flow behavior and design process

equipment. As Merrrow's studies show, the models used are far from perfect. Better

models as well as better testers are needed to advance towards reliable design. An

example of a better tester is a True Triaxial Tester (Verwijs et al., 2002) which would

allow full control of powder deformation.

Stick-slip in Powder Flow

This research investigates stick-slip in powder flow. Stick-slip is a discontinuous

flow of the powder, making it deform in steps. Stick-slip is also referred to as slip-stick.

It might be a chicken and egg discussion, but the author believes that the correct term is

stick-slip since the stick comes prior to the slip. A powder can not slip without first

sticking, while the powder does not have to slip before it sticks.

Stick-slip is a well-known phenomenon in tribology (Bhushan, 2002; Bowden and

Tabor, 1950). In that field it expresses itself in squeaking door hinges, car tires, or chalk

on a blackboard. The mechanism of stick-slip in powder flow is not known. Stick-slip can

cause problems in industry in several ways. The best-known phenomenon is silo quaking

or honking (Roberts and Wensrich, 2002). The step-wise deformation of the powder

results in violent vibrations in the silo structure, which is quaking. The empty headspace

of the silo works as a soundboard making a very loud honking sound. The honking sound

is a noise hazard while the quaking over time can cause significant structural damage.









In this research the frequency and magnitude of stick-slip events are investigated.

The frequency or time scale can be converted into a characteristic length scale. This

length scale can be used to connect the discrete particle scale with the continuous powder

scale. An understanding of the stick-slip behavior will give a better understanding of

powder flow in general. The author proposes that stick-slip occurs in all powders, but not

all powders exhibit the effects.

Focus of Dissertation

This dissertation focuses on stick-slip in powder flow. Powders are modeled as a

continuous medium for flow purposes. Of course powders consist of individual particles.

At some length scale the influence of individual particles is not noticeable in the bulk.

The border between the discrete particle scale and continuous bulk scale is not clear. It

will be different for different powders. Every powder has a certain coherence length, also

called sphere of influence. The coherence length describes how far the influence of an

individual particle reaches into the bulk. This coherence length is a key parameter in the

systems approach and will connect the particle scale with the powder scale.

Stick-slip is a phenomenon that exhibits itself on a bulk scale while it is caused by

the structure in the powder, which consists of the individual particles. Therefore, stick-

slip is an indicator of the coherence length. Hence the title of this dissertation is Stick-slip

in Powder Flow: A Quest for Coherence Length.

The aim of this work is to establish the causal links between several operating

variables and the magnitude and frequency of stick-slip events. Based on the results, a

mechanistic model is proposed which explains the origin of stick-slip. The mechanism

includes the formation of a structure of clusters in the powder. The mechanistic

understanding of stick-slip and these clusters will help the formation of powder flow









models based on first principles. In short, the roadmap of the work consist of the

following objectives:

* Understand the artifacts of conventional testers.
* Measure stick-slip as a function of powder and operating variables.
* Measure particle properties.
* Correlate particle properties with macroscopic stick-slip using simulations.
* Develop a stick-slip model.

Outline of Dissertation

The chapters is this dissertation can be read separately. The structure is such that

they form a complete work though. The next chapter, chapter 2, is an introduction to the

subject which is useful for all the consecutive chapters. The chapter describes the two

system levels that are discussed in this work, the micro scale and the macro scale, and

how these two scales can be connected. The micro scale section explains how particles

interact. The macro scale section deals with bulk powder properties. The traditional

approach to powder flow is explained and a description is given of the commercial

powder tester that is used in this research. Finally, tools are given to connect the micro

and macro scale. The main focus here is on discrete element method (DEM) modeling.

This technique models a large number of particles, which enables the connection between

the particles, the structure they form, and the macroscopic response.

For reliable data a good understanding of the test equipment is important. None of

the existing powder testers is perfect, so it is important to know what the strong and weak

points of different testers are. In chapter 3 some of the artifacts of commercial powder

testers are investigated. Stress distributions and anisotropy in testers are measured and

DEM modeling is used to explain the measured artifacts.









Chapter 4 describes a newly developed powder tester, the uniaxial tester. The tester

has the unique capability that it can deform a powder both stress controlled or strain

controlled. It is often claimed that there is no difference between those two control

systems, but this chapter shows that in the case of stick-slip it does make a difference.

The main chapter of this dissertation is chapter 5, which describes the stick-slip

research. A literature review is given of the work that is performed by others

investigating stick-slip in powders. Experimental work is presented showing the

investigation of different powder and operating parameters. Based on the results a

mechanism for stick-slip is proposed.

In chapter 6 the inter-particle forces between polystyrene particles are investigated

with the atomic force microscope. The adhesion and friction between the particles are

measured and the results are simulated with the discrete element method to validate them.

Using the correct particle interactions, a small size shear box is simulated to capture the

stick-slip behavior with the discrete element code.

The last chapter discusses the findings of the combined chapters to come to the

final conclusions. This dissertation is a small step on the road to powder flow modeling

based on first principles. In this chapter suggestions are made for future research to

further advance towards the final goal of reliable powder flow.














CHAPTER 2
POWDER TECHNOLOGY

Powder Flow

Powder mechanics is also referred to as powder- or granular flow. This term might

be confusing since it is different from the flow of fluids. Powders can transfer shearing

stresses under static conditions and have a static angle of friction greater than zero. This

enables powders to form a heap, while fluids level off. Many powders also posses a

cohesive strength after consolidation. This allows powders to form stable structures like

arches, which fluids cannot. The flow of powders is a deformation of the powder, which

can be the collapse of a stable arch. This flow is also called failure of a powder. Before a

powder fails, it dilates, which is a volume expansion. The stress level at which this

dilation occurs is called the compaction/dilation boundary.

The shear stresses during slow flow or deformation of powders are usually

independent of the shear rate and dependent on the mean pressure. This is the opposite in

most liquids, where the shearing stresses are dependent on the shear rate and not on the

mean pressure. Therefore attempts to describe granular flow with fluid mechanics have

not been successful (Jenike, 1964).

The first investigations on granular flow go back to the 18th century. Coulomb

(1773) hypothesized that soils fail along a rupture plane. Roberts (1882) reported that the

weight of granular material measured on the bottom of a bin reduced with increasing

material head. This is due to the walls, which support part of the weight. Janssen (1895)

showed this mathematically by using a continuum approach. Reynolds (1885) found that









a compacted material expands before it fails, which is known as dilantancy. Hvorslev

(1937) combined Coulombs and Reynolds results in a three dimensional stress-strain-

porosity diagram. This diagram was further developed by Roscoe et al. (1958) and is

shown in Figure 2-1. The diagram shows the failure surface of granular material as a

function of the normal and shear stress and the porosity of the material.

EYL
T

YL















Figure 2-1: Hvorslev diagram showing the effective yield locus (EYL) and a yield locus
(YL) which are the projection of the critical state line (C1C2) and a failure
surface respectively. The space is spanned by the normal stress (c), the shear
stress (c), and the porosity (s).

The Hvorslev surface is divided into two regimes by the critical state line (C 1C2).

At the left side the material fails under shear while at the right side the material

consolidates. The three dimensional surface can be projected onto the two dimensional

shear stress- normal stress space. The critical state line projects as the effective yield

locus and the yield surface projects as a series of yield loci, each corresponding to some

initial consolidation (see Figure 2-2). Jenike (1961) developed an engineering approach









to design silos based on powder flow characteristics. Jenike's method is still being used

to design silos.



T effective yield locus


yield loci


C




Figure 2-2: Family of yield loci, with the cohesion c and angle of friction q given for the
top locus.

Macroscopic Scale

Powder Mechanics

The flow behavior of powders is very similar to that of soils. Since much of the soil

mechanics research advances that of powder mechanics, much can be learnt from civil

engineers. The tester Jenike developed originates from the shear box used in soil

mechanics. There are some major differences though. The stresses exerted on soils are

usually one to three orders of magnitude higher than the stresses powders experience. The

deformation of soils is usually not as large as the deformation of a powder. Soils are

often saturated with an interstitial fluid, while powders are generally dry. The scope of

the research in soil mechanics differs from that in powder technology. In soil mechanics

one does not want flow to occur (since that might mean the collapse of a dike) while in

powder technology flow is generally wanted. Civil engineers are more concerned about

how granular material approaches failure, i.e., the region before the compaction/dilation









limit. In powder mechanics the interest is the region between the compaction/dilation

limit and failure. Those factors prevent a direct comparison between soil mechanics and

powder mechanics.

In powder mechanics a continuum approach is used to describe the state of a

powder. This means that the particulate structure and voids are disregarded. If a sample

volume is taken infinitesimally small, it will still behave similarly as the bulk and its

properties are described by continuous functions. It allows the definition of stress at a

point. For the continuity principle to hold, the sample size of powders measured should

be larger than the minimum sample size. The coherence length, i.e., the length in which

the effect of individual particles vanishes, also describes this minimum sample size. It is

the smallest volume that still behaves as a bulk powder. The minimum powder sample is

generally not known and is investigated is this project.

Besides the continuity assumption, two other assumptions are often made in

continuum mechanics, homogeneity and isotropy. Homogeneity means that the material

has identical properties at all points in the material. Isotropy means that the material

properties are the same in all directions. In powder mechanics those assumptions can

often not be made. Powder density can for example vary with the bed height, which

means that the sample is not homogeneous. After deposition of flaky particles the ability

to sustain stress in the vertical direction can be different from that in the horizontal

directions, which means the sample is not isotropic.

Stress (o-) is defined as a force (F,) in direction that acts on an area (A,) with

normal vector i, as shown in equation 2-1. A distinction can be made between normal









stresses (o-,), where the force works on the plane i in the direction i and shear stresses (o,

or zr), where the force works on the plane i in the direction.


a, = lim 2-1


The equilibrium stresses on a cubical powder sample can be described by a stress

tensor. This is shown in Figure 2-3, where the normal stresses are on the diagonal and the

shear stresses fill the rest of the stress tensor. The tensor can be resolved such that all the

shear stresses are zero. The resulting normal stresses are called the principal stresses. In

order of increasing magnitude, these stresses are called the minor, intermediate, and

major principal stress and denoted by 0-3, -2, r-, respectively. In powder mechanics

compressive stresses are positive and tensile stresses are negative.

C11

13 23 3 U11 T12 T13


1 4 33` T21 22 T23
3 2 21 31 T32 33


Figure 2-3: Stresses on a powder sample with the corresponding stress tensor.

The three principal stresses are the eigenvalues of the stress tensor and form the

real roots (A) of the so-called characteristic equation 2-2. The eigenvectors of the tensor

denote the direction of the principal stresses. I], 12, and 13 are the invariants of the

equation and are shown in equations 2-3 to 2-5. The invariants are independent of the

rotation of the coordinate system.

A' I A + -I = 0 2-2

I =trS )= CJ +C2, +C3 2-3


2-3










2 =det 11 12 + det 22 23 + det 11 13 2-4
21 22 32 33 31 33

13 = detL() 2-5

A form of the first invariant that is commonly used is the mean stress (o0,) or

hydrostatic stress (oH), which is the normal mean of the three principal stresses. Some

models use the stress deviator or deviatoric stress tensor S', equation 2-6, where I is the

unit matrix. The directions of the principal deviatoric stresses and principal stresses are

the same. The values of the principal deviatoric stresses are oU om. The invariants for the

deviatoric stress can be calculated similarly to the invariants of the stress. The first

invariant of the deviatoric stress will be equal to zero. In rock mechanics it is believed

that the deviatoric stress causes change of sample shape and the hydrostatic or mean

stress causes volumetric change. For granular material this is not totally correct. A

granular material can dilate when it is sheared with a deviatoric stress.

= S 1 2-6

Similar to the stress tensor, there is also a strain tensor, describing the state of strain

or deformation of a powder sample. There are several ways to define strain. In this

research the Cauchy definition is used, which is the change in length per unit of initial

length. Figure 2-4 shows a schematic representation of the normal strain and the shear

strain. Equations 2-7 and 2-8 show the calculation of the strain components. It can be

seen that contractive strains are positive and extensive strains are negative, which is

common in powder mechanics. Beside the normal and shear strains, a sample can also

strain by rotation. Models often use the strain rate, which is the derivative of the strain

with respect to time.






16


A3

All 1 F 712 713

11 1 = 721 -22 723

2Y3 732 133
13 A13

Figure 2-4: Schematic of deformation with corresponding strain tensor.

El A 2-7



2 = + 2-8


Constitutive Models

To be able to predict powder flow constitutive equations have to be developed. It is

difficult though to come up with a universal equation for powders. Some powders behave

plastically, others elastically, some linearly but most not. Therefore, using the continuum

mechanics approach, several models have been developed. Jenike (1961) used an ideal

Mohr-Coulomb model of friction to describe powder flow. The Coulomb yield criterion

describes plastic deformation in the shear plane. The model assumes that the shear stress

rin a failure plane is independent of strain or strain rate, but dependents on the normal

stress o on the shear plane as represented in Figure 2-2. The yield loci can be described

by equation 2-9, where pu is the friction coefficient, q the angle of friction, and c the

cohesion. The cohesion is a function of the compaction or "history" of the powder. The

failure (or strength) of a powder is therefore dependent on that history. Jenike formulated

a way to measure and calculate this strength, which is called the unconfined yield

strengthfc. The plot of this unconfined yield strength as a function of the pre-shear stress









is called the flow function. This model is mainly used in engineering to calculate the

limiting state of stress in powders, e.g., during failure. The Mohr-Coulomb criterion can

usually predict the compaction/dilation limit, but can not predict the full stress-strain

curve.

r = / + c = tan(b) + c 2-9

The von Mises model predicts that yield occurs when the value of the second

deviatoric stress invariant is equal to a constant. This constant is a material property that

indicates the onset of yield. The deviatoric stress does not include the hydrostatic stress.

It is known that the compaction of the powder does influence the strength of a powder.

Therefore Drucker and Prager (1952) included the first invariant of the stress tensor.

Equation 2-10 shows the Drucker-Prager model where, 12 is the second invariant of the

deviatoric stress, I1 the first invariant of the stress, aDP a frictional coefficient, and kDp a

cohesive coefficient.

= aDP1 + kDP 2-10

The model assumes plastic deformation when the stress state of the powder is at the

failure surface. Most powders deform before that point, which is called strain hardening.

Powders are better described with an elasto-plastic strain-hardening model. Drucker et al.

(1957) introduced a hardening cap to their model which accounts for the volumetric

plastic deformation before failure. The change of bulk powder density before failure

dilatancyy) is used in the model.

Further development of models was through the Cam-clay model and the modified

Cam-clay model. The Cam-clay model assumed a bullet-shaped cap, while the modified









Cam-clay model assumes an elliptical shape. The elliptical model is given in equation 2-

11, where Mis a material parameter and ao is a strain hardening parameter.


12 ,= 2-11


All models described above rely on the first two invariants of the stress or stress

deviator. It is well known that in the failure of geomaterials the third invariant is also

involved. This is shown in the right figure in Figure 2-5 by the conical shape of the

failure surface. If the shape of the deviatoric plane (distance c from the origin) would be

circular, it could be described by the second invariant of the deviator and the mean stress.

A point P on the surface can be found using the second and third invariant of the deviator,

as shown in equation 2-12.


0C












y--------------- C2




Figure 2-5: Possible shapes of the failure surface; hexagonal (left) and triangular (right).


cos3a= 2-12
l2j2









For many geomaterials the failure surface is not circular but triangular with

rounded comers as shown in the figure. For powders the failure surface has a hexagonal

shape. The corners of the hexagon might be rounded as well. To describe the roundness

of the corners, the third invariant of the deviator is needed. To calculate the third

invariant the full stress tensor is needed. The full stress tensor can only be measured with

a true triaxial tester.

Powder Structure

The current flow models assume that the powders are homogenous and isotropic.

They do not take the structure of powders in consideration. Especially cohesive powders

are known to be non-homogenous. They form agglomerates and clusters with different

porosities. The structure that is formed in this way is difficult to describe, especially since

porosity is a scalar parameter (Scarlett et al., 1998).

The isotropic assumption is a simplification of reality as well. It has been shown

that most powders have a tendency to show anisotropic behavior. Following Wong and

Arthur (1985), there are two different forms of anisotropy, namely inherent and induced.

Inherent anisotropy is the result of the deposition process and the particle characteristics.

Therefore it is also called structural anisotropy. It is found that after raining the particles

in a tester, the contact planes between particles were oriented mostly horizontally due to

the particle's own weight (Feda, 1982). The degree of structural anisotropy will vary with

different deposition processes and particle characteristics. Particles that are elongated or

flake-like will show more anisotropy than spherical particles.

Induced anisotropy, also called mechanical anisotropy, is due to the strain

associated with an applied stress. According to Li and Puri (1996), the mechanical

anisotropy is the result of structural anisotropy. This is not true, because a powder









existing of spherical particles can also be anisotropic (Zhupanska et al., 2003). Induced

anisotropy can be significant. The effect of anisotropy is most clearly shown in the angle

of internal friction and the unconfined yield strength, which can change by 10% (Arthur

and Menzies, 1972) and 200% (Saraber et al., 1991), respectively.

Although a distinction is made between inherent and induced anisotropy they are

actually the same. Inherent anisotropy is induced by the deposition process, which is a

mechanical process. So inherent anisotropy is a form of induced anisotropy.

Powder Testers

An extensive comparison between powder testers is given by Schwedes (2003).

Powder testers can be divided into direct shear testers and indirect shear testers. In direct

shear testers the location of the shear plane or zone is defined by the geometry of the

tester. They are based on the shear box model and examples are the Jenike shear cell, the

Peschl cell, and the Schulze cell. These testers are able to measure the flow function,

following a precisely described procedure (ASTM, 2000, 2002; EFCE, 1989).

The results from different testers vary due to the different geometries of the sample

holders. The exact stresses and strains in direct shear testers are not known (Bilgili et al.,

2004; Janssen, 2001). It is assumed that there is a uniform stress and strain in the testers,

but that is not the case. The control of direct testers is limited. In general only the normal

stress, and shear strain rate can be controlled and only the shear stress can be measured.

Therefore, direct shear testers are not suitable for fundamental research towards general

constitutive laws.

Indirect shear testers do not force a shear zone in the powder at a specific location,

but deform the powder as a whole. The powder will form shear zones where it needs to

and this should be independent of the geometry of the tester when the sample size is large









enough. Most indirect shear testers are biaxial cells. These testers have more control than

the direct shear cells. In general two principal stresses or strain rates can be controlled

and the third principal stress is measured. An example of a biaxial tester is the Flexible

Wall Biaxial Tester which is discussed in chapter 3.

Civil engineers use a cylindrical tester, which is called the von Karman tester. They

claim that this tester is a triaxial tester. It is assumed that the hoop stress is equal to the

radial stress, so those two principal stresses cannot be controlled independently. It is very

difficult to use this tester for powders since the tester cannot measure in the low stress

region that is important for powder flow (Verwijs et al., 2003). It is not certain whether

the whole powder sample is in deformation. To get a useful strain tensor, the whole

powder sample has to be in deformation. For this, knowledge of the minimum sample

length is important. The von Karman tester is not suitable for the fundamental research

conducted in this work. Figure 2-6 shows the operating window for stress control of a

biaxial and the von Karman Tester. It can be seen that both testers cannot be controlled in

the full three-dimensional stress regime. Therefore a true triaxial tester is needed.



II III 3)









Figure 2-6: Operating window for stress control of a biaxial (right) and von Karman
tester (left).

For soil mechanics a few true triaxial testers exist. One is located at the university

of Bristol (Airey and Wood, 1988). This tester is not suitable for powder flow research









though. The stress regime the tester is designed for is orders of magnitude higher than the

stresses occurring in powder flow. The robust design necessary to accommodate these

stresses is far from frictionless. The friction in the system is negligible for the stresses

applied in soil mechanics, but is higher than the stresses that are used in powder flow.

Therefore, measurements in this low stress regime will be unreliable.

Direct Shear Cell Measurements

The most widely used type of powder tester is the direct shear cell. The best known

are the Jenike shear tester, the Schulze shear tester, and the Peschl tester. The tester used

in this research is the Schulze tester. The test procedure is very similar to other direct

shear testers, especially the data reduction. The Schulze tester is an angular direct shear

cell that allows the determination of the unconfined yield strength as well as the internal

angle of friction and the effective angle of internal friction. The advantage of the Schulze

tester is that it has unlimited travel. The disadvantage is that the tester shears the powder

in an angular direction, and not linearly as in the Jenike cell. Figure 2-7 shows a

schematic of the Schulze tester.

F, 111-


Figure 2-7: Schematic of the Schulze tester.









A powder sample is placed in the base (which has a rough bottom) that is rotating.

A lid with vanes is placed on top of the powder, such that the vanes stick into the powder.

The top lid is held in place by tension bars, preventing it from rotating. The bars are

connected to load cells which measure the force that is necessary to keep the top in place.

Through a hanger the top lid can be loaded with weights to compact the powder with a

normal stress. Since the bottom is rotating and the top is stationary, the powder will shear

somewhere between the bottom and the vanes. The load cells measure this shear force.

The standard method to measure the unconfined yield strength of a powder is a

steady-state deformation under a certain normal load, followed by a failure with a lower

load. The procedure is described in an ASTM standard (2002). For normal experiments

the powder is deformed with a constant clockwise angular velocity until the shear stress

reaches a constant value. This step is called the pre-shear and at this point there is no

change of density and the powder is in steady state deformation. The state of stress of the

powder at this point is the critical state. Once this state is reached for a certain normal

load, the deformation is stopped and the normal load reduced. With this reduced load, the

shearing is started again to fail the powder. This sequence is repeated several times with

the same load during steady state deformation but different failure loads. In the shear

stress-normal stress space these failure points form the yield locus corresponding to the

chosen critical state of the powder. From the yield locus the unconfined yield locus can

be found by extrapolating a Mohr circle through the origin and tangent to the yield locus

(see Figure 2-8). The major principal stress of this Mohr circle is the unconfined yield

stressfc. The major principal stress of the Mohr circle through the steady state point and

tangent to the yield locus is called the consolidation stress o-. The combination of the









consolidation stress and unconfined yield stress for several yield loci will form the flow

function.



-4






__f I

Normal Stress (kPa)

Figure 2-8: Diagram showing the yield locus (solid line), steady state point (diamond),
failure points (triangles), major Mohr circle (dashed semicircle), and unconfined Mohr
circle (dotted semicircle).
Microscopic Scale

ined Although the focus in the powder flow field has been on the macroscopic scale for

many years, more and more researchers are investigating the microscopic particle scale to

explain macroscopic phenomena. As explained in the previous chapter, the macroscopic

scale is built up from the microscopic scale. A good understanding of the processes on

the microscopic scale will enable the explanation of macroscopic phenomena.

Pressure Mapping

It is generally known that the stresses that are measured with conventional powder

testers are average stress values. In powder testing, the overall normal force on the

powder is divided by the area the force acts on to give the normal stress acting on the top

of the powder sample. This normal stress is assumed equal to the normal stress acting

throughout the material. This is not correct since the walls of the tester will carry part of

the load. Because of this, there will be a stress distribution in both the radial and axial

directions within the powder rather than one overall stress.


th the radial and axial

directions within the powder rather than one overall stress.









Within the measured area there can be large deviations in the local stress. These

deviations lie in between the microscopic and macroscopic scale. The stress deviations

can be caused by individual particles pressing against the wall forming hot spots, which

is a microscopic phenomenon. The deviations can also be caused by the structure of the

powder, which is in between the two length scales.

In this work the Tekscan mapping technique (Hunston, 2002) is used to measure

the stress distributions on surfaces. The Tekscan sensor consists of 1936 sensels (pressure

sensor points) arranged in a square grid configuration (see Figure 2-9). The sensor pad

has a resolution of 15.5 sensels per cm2. Experiments conducted with these pads revealed

that pressure measurements are accurate to within 10% of the measured stress when used

in powder shear conditions (Bilgili et al., 2004; Johanson and Bucklin, 2004).







...... .. Conductive 'Substrate
No Leads










or dielectic)

Figure 2-9: Schematic of a Tekscan sensor pad.

Inter-Particle Forces

Forces of different origin can exist between particles. The most common forces are

the van der Waals forces, electrostatic forces, electromagnetic forces, and capillary









forces. These forces have a physical origin and are called long-range forces. Short-range

forces are interactions involving covalent bonds and usually act up to a distance of 0.1 -

0.2 nm. The long-range forces act beyond this point up to several nanometers. Not all

long-range forces have to be present or significant in a particle system. The dominant

forces in this research are the van der Waals forces and capillary forces.

The van der Waals forces can be subdivided into the London dispersion force, the

Keesom force, and the Debye force. The London dispersion force is an induced dipole

interaction between atoms and molecules, created by fluctuations of electronic charges.

The Keesom attraction force is an interaction between rotating permanent dipoles. The

Debye force is an interaction between rotating permanent dipoles and polarizable atoms

and molecules. The London force is usually the dominant force since all materials are

polarizable, while for the other van der Waals forces permanent dipoles are required. A

detailed description of the van der Waals forces can be found in several books (Bhushan,

1999; Israelachvili, 1985).

For atoms, the London dispersion potential is combined with the Born repulsion

potential in equation 2-13, which is called the Lennard-Jones Potential W, with B the

Born repulsion constant, C the London dispersion constant, and D the separation of the

atoms. The Born repulsion is due to overlap of electron clouds of two bodies when they

approach within 2 to 3 A.

B C
W = 2-13
D12 D6

On a particle scale, when the separation distance D between two identical spheres is

small compared to their radius R, the van der Waals interaction forces FvdW can be

approximated with equation 2-14, where A is the Hamaker constant. The van der Waals









forces between bodies that are not spherical or of dissimilar size can be calculated using

their specific geometries. The result for the interaction between two spheres of different

size and the interaction of a sphere with an infinite plate are given in equation 2-15 and 2-

16, respectively.

AR
Two identical spheres: Fd = 2-14
12D2

A RR
Two spheres of different radius: FdW = 2 RR2-15
6D (R, + R2)

AR
Sphere with plate: FAdw 2-16
6D

Derjaguin proposed a method to approximate the interaction between surfaces

based on the interaction free energy per unit area. Using a slice method the adhesion

force for different geometries can be calculated. The adhesion force between a sphere

with radius R and a plate using Derjaguin's approximation is calculated using equation 2-

17, where W123 is the work of adhesion between two plates of species 1 and 3, with

species 2 in between.

Fadh = 2zRW123 2-17

Electrostatic forces generally act across a longer range than van der Waals forces.

In wet systems the surface can partially dissociate and form an electrostatic double layer.

This double layer causes a repulsive electrostatic force between surfaces. In dry powders

dissociation of the surface is not very likely. For non-conducting or poor-conducting

materials it is possible though to accumulate surface charges. A common manner to

acquire these surface charges is through pneumatic conveying of powders. The

electrostatic interaction between particles can be attractive or repulsive. Figure 2-10













shows a typical repulsive electrostatic force between two bodies as well as the London

force and Born repulsion as a function of the distance between the bodies.

I.

iI
ii 4


I 1


0





13 /
S1. Born repulsion
2. van der Waals attraction
I 3. Born repulsion & van der Waals attraction
I 4. Electrostatic repulsion
\I 12 5. All interactions combined
I

Separation Distance (nm)

Figure 2-10: Force-distance relationships for the Born repulsion, London van der Waals
attraction, electrostatic repulsion, and combinations of those.

An additional force that can exist between particles is a capillary force. A liquid

bridge can form between particles even when a powder is relatively dry. An extensive

review of inter-particle capillary forces is given by Esayanur (2005). In his work

Esayanur connected capillary forces with powder cohesion. Rabinovich et al. (2000a;

2000b) developed a simplified relationship for a capillary adhesion force Fad between a

smooth spherical particle and a flat substrate with nanoscale roughness. The relationship

is given in equation 2-18, where yis the surface tension of the liquid, R the radius of the

adhering particle, Hap the maximum height of the asperities above the average surface

plane, r the lesser radius of meniscus, and cos0 = (cosOp + cos0s)/2. The angles Op and Os


ius of meniscus, and cosO = (cosOp + cosOs)/2. The angles Op and Os









are the contact angles of the liquid on the adhering particle and substrate, respectively, as

shown in Figure 2-11. Equation 2-18 is applicable only when the meniscus is large

enough to span the distance between the adhering particle and the average surface plane

and when the radius of the meniscus is small compared to the adhering particle.

H
F,d = -47ryRcos 0 1-- 2-18
2rcos0


R








Hasp




Figure 2-11: Schematic representation of a liquid bridge between a sphere and a rough
plate (with permission, from Esayanur (2005)).

Contact Mechanics

When considering a powder bed on a particle scale, besides the interaction forces,

the particle contact behavior is of eminent importance. When there is no significant

plastic deformation, the contact area depends on the elastic response of the material. In

1888 Hertz developed a contact mechanics theory between an elastic spherical body and

a plate. His work forms the basis of several theories. In his theory Hertz assumed no

inter-particle forces, so a contact deformation occurs due to an external compression

force F. When a particle and a plate are pressed together, they form a circular contact

area. The radius a of the contact area follows a relationship as shown in equation 2-19,









where R is the particle radius and K the reduced elastic modulus. The reduced elastic

modulus can be calculated from the Poisson's ratio v and the elastic modulus E of the

two materials using equation 2-20. The contact deformation 3, shown in Figure 2-12, can

be calculated using equation 2-21.


a= 2-19
K )


K1 3 1 ~pl~ l-v, 1,, ere
h 2-20
K 4 pate phere


2a
83- 2-21
R

In the 1970's two famous theories were developed from Hertz's work; DMT

mechanics and JKR mechanics. The DMT theory is developed by Derjaguin, Muller and

Toporov (1975) and includes long-range attraction forces to Hertz's theory as shown in

equation 2-22, where Fadh is calculated using equation 2-17. The contact deformation is

Hertzian, so calculated using equation 2-21. The model shows no hysteresis between

loading and unloading, thus excluding the possibility of pull-off forces. The applicability

of the model is limited to stiff systems with low adhesion forces.

r
a= L(F +F )d 3 2-22
K

The JKR theory is developed by Johnson, Kendall and Roberts (1971) and does not

include the long-range forces outside the contact area but does include short-range forces

inside the contact area. The radius of the contact area is given in equation 2-23. The

model accounts for adhesion hysteresis, forming a neck at unloading. Therefore, the









deformation is not Hertzian but according to equation 2-24. The JKR model is suitable

for materials with a large adhesion force and large radius.

a=[_ ____ X
a (F + 3rW123R + 6W23RF + (3'TW1R)23 2-23


= 2 6W3 2-24
R 3 K

F






JKR neck







a Hertz ajKR

Figure 2-12: Schematic representation of Hertzian and JKR contact radius and
deformation.

In the 1990's two additional theories were developed; BCP mechanics and Maugis

mechanics. The BCP model is an empirical model, developed after years of force curve

studies. The Maugis theory is a complex model, which applies to all systems and

connects the region between the DMT and JKR theory. An extensive comparison

between the different contact models is given by Burnham and Kulik (1999).

The friction between particles is very important for powder flow evaluation. When

a powder is sheared, particles have to slide along each other and the friction between their

surfaces will define the ease with which that is accomplished. This inter-particle friction









is thus a major contributor to the shear force. In the macroscopic world the friction of a

material is defined by the friction coefficient. The friction coefficient is the ratio of the

friction force at which a body starts moving over the normal force pressing the bodies

together. Since the friction force generally is a linear function of the normal force, the

friction coefficient is a constant. The value can range from near zero for frictionless

materials to up to 4 for rubber and some metals. Most common materials will have a

friction coefficient between 0.1 and 0.7 though. There can be a different coefficient for

static and kinetic friction, where the first is slightly larger than the latter.

Friction is proportional to the contact area of the contacting bodies, as shown in

equation 2-25 (Adams et al., 1987). It is not the macroscopic area that is of importance,

but the actual area of contact points A. Due to surface roughness the actual contact area is

much smaller than the macroscopic area of two bodies. When two particles are sliding,

adhesive junctions are formed and work is required to rupture these junctions. The

strength of these junctions is represented by the interface shear strength T.

Ffrcon = rA 2-25

The contact area will increase with increasing normal load due to deformation of

the contact points. When purely plastic contact deformation occurs, the contact area will

increase linearly with the normal load since plastic yield is the quotient of the yield force

and the area. When an elastic Hertzian deformation of the contacts occurs, the contact

area will be a non-linear function of the normal load. From equation 2-17 it can be seen

that the contact area is proportional to the normal force to the power two-thirds. This is

why the friction force can be expressed with the empirical equation 2-26, where k is the









friction factor, F the normal force, and n the load index with a value between 2/3 and 1.

When n is unity the friction factor becomes the friction coefficient as discussed before.

Ffrchon = kF" 2-26
frichon

Adams et al. (1987) combined Hertz mechanics with equation 2-25 to form

equation 2-27. They separated rin an intrinsic shear strength at zero load to and a

pressure coefficient ato account for the pressure dependence of r. Using either the DMT

or JKR model, adhesion can de added to the normal force in equation 2-26 and 2-27.

(RF %
Ffr = c ronO +aF 2-27
SK

Discrete Element Modeling

Discrete element method (DEM) modeling is generally used to investigate the

effects of microscopic mechanical properties (rheological properties of particles, friction

between particles and the wall) on the macroscopic mechanical response of powder

materials. The critical parameters of particle packing and structure in an assembly are the

changes in contact orientations (contact normals) of particles, particle arrangement (i.e.,

coordination number and distribution of neighbors), and structural porosity due to

specifically applied loading paths. These changes in microstructure are mostly

responsible for directional anisotropic behavior of powders. Using discrete element

analysis, the evolution of a stress-strain response of a powder as well as the evolution of

microstructure due to different loading histories can be simulated.

DEM was introduced by Cundall in 1971. He considered particulate material as an

assembly of particles, which are in direct interaction to each other. The program was

developed to simulate the behavior of rock by masses that interact through springs and









dashpots. Cundall and Strack (1979) developed the program to simulate the quasi-static

shear deformations of soils. DEM computational procedures alternate between the

application of Newton's second law of motion on particles and their contact interaction.

Contact mechanics are used to define the force between particles that are in contact. For

every particle the resultant force is calculated from all its contacts and additional forces

like gravitational forces. Newton's second law is used to calculate the acceleration of the

particle due to its resultant force. All particles are allowed to displace for a very short

time step using their specific velocity and acceleration. The next step all contacts are re-

evaluated and the cycle repeats.

Fabric Tensors

In many cases continuum mechanics cannot describe powder flow behavior well. In

those instances the influence of individual particles or particle ensembles is too large.

DEM simulations are a good alternative to model those situations. DEM can be used to

define the structure in the powder, e.g., anisotropy. The microstructure can be described

using the concept of fabric tensors. This concept is generally accepted in the field of soil

mechanics. These tensors are related to the spatial distribution of the particles, their sizes

and orientations, and, in particular, to the distribution of the contact normals and contact

areas, as well as other geometrical entities. This is a useful way to investigate anisotropy

since it will give the magnitude of a parameter or property as a function of the direction.

Following Kanatani (1984) the 2D invariant formulation of the distribution density

function E(n) of the unit vector n is given by equation 2-28. The fabric tensors of the

distribution, J, and Jykl, are shown in equation 2-29 and 2-30, respectively.


E(n) = ( + J, n, + Jkl nn +...) 2-28
2;ki









J = 4L(nnJ) 2-29


J 1k =16 nn nknn/ n,)n 6 ki 2-30

An example of a distribution is represented in figure 2-13 for two dimensions,

where 0 is the orientation of n measured from the vertical and Kcan be a parameter like

the stress.






K(%)


Figure 2-13: Angular distribution of parameter K, near isotropic (left) and anisotropic
(right).

For this 2D case, the density distribution function can be written as a function of

the angle 0 as shown in equation 2-31. The coefficients A, B, C, and D are a function of

the angle 0 and are given in equation 2-32. Usually, the second order tensor in expression

2-29 is sufficient to describe the microstructure of the material, but for highly anisotropic

microstructure the fourth order tensor in equation 2-30 is important.

E(n) =- +1 [2Asin0+(2C-1)cos20] +
2-31
1[(1- 8C + 8D)cos40 + 4(2B A)sin40] +..


A= (sin0cos0) C= (cos2 2-32
B=(sin0cos30) D=(cos4 )

The distribution of the intensity of the contact forces is important as well. In

cohesive powders, particles tend to form clusters. Anisotropy in macroscopic mechanical






36


response is not due to the general trends in orientation of all particles, but just to the

orientation of particles that are at the load carrying, cluster-to-cluster interfaces.

The particle size distribution also has to be taken into account. A broader

distribution in particle size leads to a more heterogeneous powder sample and,

consequently, to more anisotropic behavior. In terms of micromechanical properties, the

friction and particle shape are critical as well.














CHAPTER 3
INHOMOGENEITY AND ANISOTROPY IN SHEAR TESTERS

This chapter discusses some known (but often ignored) as well as some new

artifacts of shear cells. The intention is not to discredit these cells but to make users

aware of these phenomena in powder testers. When the existence and cause of these

artifacts is understood, it makes the interpretation of shear data much more reliable.

Effective Yield Locus

One of the most common measurements with a direct shear cell is the effective

yield locus (EYL). Therefore, it is strange that there are several definitions of this locus.

The original locus as defined by Jenike (1964) is a locus that starts at the origin and ends

at the largest Mohr circle (see Figure 2-2). Another proposed EYL goes through the

steady state deformation points of a tests series, making these points the tangent points of

the locus and the largest Mohr circles. A third variation that can be found connects the

tops of the Mohr circles, creating a locus of maximum shear stresses during steady state

deformation. The common factor between these different EYL is that they are

extrapolated to start at the origin.

Considering the projection of the critical state line of the Hvorslev diagram (see

Figure 2-1), it cannot be concluded that the intercept has to be zero since the critical state

line is asymptotic. The assumption that the EYL has no intercept comes from the soils

mechanics field. In this field the stresses of interest are very high, making a possible

small intercept insignificant. The stress levels in powders can be such that this intercept is










significant. Cohesive powders generally show critical state Mohr circles as shown in

Figure 3-1.

5 -
S= *tan(5r)+ ci

Sr = 38.60
4- (r ci=0.315 kPa


3



2



1-




0 1 2 3 4 5 6 7 8
c (kPa)

Figure 3-1: Critical Mohr circles with tangent effective yield locus for BCR Limestone,
Measured with a Schulze Cell.

The data are from experiments that are conducted with the Schulze tester with BCR

limestone. It can clearly be seen that the best fit for the EYL has an intercept with the y-

axis. This has been explained by Molerus (1978), who attributed this intercept to an

inherent adhesion between the particles. Therefore, this intercept will be called the

inherent cohesion of the powder. The cohesion between the particles is related to the

actual cohesion value as measured by direct shear testing as shown in Figure 3-2. In this

figure, the EYL for polystyrene powder is shown for different moisture contents. As

discussed in chapter 2, the moisture content can change the cohesion between the

particles due to the formation of liquid bridges between the particles. Figure 3-2 shows











that this increased cohesion between the particles causes an increase in the inherent

cohesion of the powder.

10 7


0 5 10 15 20 25
sigma (kPa)

Figure 3-2: Effective yield loci of 40 tm polystyrene powder for different powder
moisture contents, measured with a Schulze cell.

The inherent cohesion can also be obtained by data represented in a typical flow

function. This can be best seen by noting that Figures 3-1 and 3-2 show all of the

termination stress state Mohr circles for the family of yield loci that are used to generate

the flow function. As stated above, these stress state Mohr circles are tangent to the real

effective yield locus. There exists a Mohr circle stress state that is tangent to this real

effective yield locus and is coincident with a particular unconfined stress state Mohr









circle. Figure 3-3 describes this unique condition. The intercept for the real effective

yield locus or inherent cohesion c, can be related to this critical unconfined yield strength

fc* as indicated in Equation (3-1).

Real Effective Yield Locus


Yield Locus








Figure 3-3: Schematic representation of the critical strength (fc*) of a powder.


f (1- sin),
c 3-1
2 cosr

This unique cohesion value is for the case where the major principal consolidation

stress equals the unconfined yield strength. It should be noted that this condition can not

be measured by a direct shear test method and must be obtained from extrapolation of the

flow function. The flow function represents the relationship between the major principal

stress and the unconfined yield strength. In fact, the intersection of a line passing through

the point (0,0) having a slope of 1.0 intersects the flow function at this critical yield stress

fc* (see Figure 3-4). Therefore, the shear stress intercept on the real yield locus can be

found through an analysis similar to the Jenike arching analysis. This implies the curious

result that stress states during steady flow depend on the cohesive nature of the material,

implying that incipient failure conditions of a bulk material influence flow modes where

continual deformation occurs. Such a result could explain why solids flow rates are









sometimes influenced by the cohesive nature of the material. It may also explain why

channeling occurs in fluidized beds with cohesive material. In fluid beds, inherent

cohesion may play a major role in the flow profile.

fc
Flow Function

{U, f*) / I





/\ Slope = 1



Figure 3-4: Method of computing critical strength from flow function to be used for
o inherent cohesion calculation.


Since the Jenike procedure forces the EYL through the origin, that procedure will

produce a different effective angle of internal friction for every stress state as shown in

Figure 3-5. The angle is found by taking the angle of the locus that goes through the

origin and is tangent to an individual Mohr circle. Since both the EYL according to

Jenike and the real EYL are tangent to a Mohr circle with major principal stress o-, it can

be shown that the relationship between the effective angle of internal friction as defined

by Jenike 6S, the real angle of internal friction r, the inherent cohesion c,, and this major

principal stress is given by equation 3-2. This equation is used for the dotted curve in

Figure 3-5.


sin os sin 3-2
c, cos a1


cos ( a1









90


80


70



601


50 s


40



0 11.3 2 3 4 5 6 7 8 9 10
a-1 (kPa)

Figure 3-5: Effective angle of internal friction according to Jenike (5s) as a function of the
major principal stress (o-1) for BCR limestone, measured with a Schulze Cell
(diamonds) and according to equation 3-2 (dotted line).

The smallest Mohr circle, which has a minor principle stress of zero and some

positive value for the major principle stress, denotes the limit of the curve and will have

an effective angle of internal friction (Sj) of 900. The major principal stress value of this

limiting Mohr circle is 1.3 kPa for the measured BCR limestone. From equation 3-2 and

Figure 3-5, it can be seen that for large values of the major principal stress both effective

angles of internal friction converge. As stated, that is why civil engineers are not

concerned about the difference between the two angles. In powder technology, with

operating pressures between 0 and 50 kPa, the real effective yield locus (with a possible

intercept) should be used as opposed to other mentioned loci.


entioned loci.









Flow Function

As mentioned in chapter 2, the flow function of a powder is very important for

process equipment design. It is considered an inherent powder property, which is known

to be dependent on temperature, moisture, particle size distribution and other factors. The

flow function defines the unconfined yield strength (fe) of a powder as a function of the

history of the powder. This history is described by the major principal stress during pre-

shear. This section shows the influence of moisture on the strength as well as the

directional dependence of the strength. This directional dependence in direct shear cells,

or anisotropy, is partly explained as a geometric artifact of the testers and is called

geometric anisotropy.

Influence of Moisture

As discussed in chapter 2 and the last section, the cohesion of a powder increases

with increasing moisture content. The effective yield data shown in the previous section

are from a series of data that is used to measure the strength of a polystyrene powder as a

function of the moisture content. The flow functions are shown in Figure 3-6. It can be

seen that the flow functions initially increase with increasing moisture content, but after a

moisture content of 0.5 % decrease slightly. This has been shown by Esayanur (2005) and

is explained as follows. Initially, the moisture can form liquid bridges between the

particles. When the moisture content increases up to a point where the water can form a

monolayer around the particles, the formation of liquid bridges ceases. The monolayer of

water will act as a lubricant, decreasing the friction between the particles, hence lowering

the strength.







44


7

-0-0.6%
6 --*-0.5%
-U- 0.4%
-"--0.3%
5 -"- 0.2%


4-
a_

S3-


2


1-i



0 5 10 15 20 25 30
0-1 (kPa)


Figure 3-6: Flow function of polystyrene powder for different moisture contents,
measured with a Schulze cell. Connecting lines are for visual clearance.

Directional Dependence

As discussed in chapter 2, many researchers have shown that a powder can show

directional dependence or anisotropy in certain properties. For the strength of a powder

this means that the flow function has different values dependent on the direction in which

it is measured. This has been reported with several testers, but no literature has been

found on anisotropic strength measurements with the Schulze shear cell. To measure

anisotropy in a Schulze cell certified BCR limestone was used. The results for a standard

strength test were compared with Akers (1992), which is the international standard, and

Saraber et al. (1991). As can be seen in Figure 3-7, the results are alike considering the

accuracy of the testers.









5 -
*Akers Saraber AVerwijs







4 A


2- A
A

A
1 -




0 2 4 6 8 10 12 14
1 (kPa)

Figure 3-7: Comparison of strength measurements of BCR limestone with the Schulze
cell (PERC) with Akers (1992) and Saraber et al. (1991).

Results from the standard procedure were compared with an anisotropic procedure.

With the anisotropic procedure the direction of failure was reversed from clockwise to

counterclockwise. To do this, tension rods were designed that can be used under

compression. Since the load cells of the Schulze cells are to be used both in compression

and tension a calibration graph was produced for both directions (see appendix A).

The results of the experiments are shown in Figure 3-8. It can be seen that the

strength of the powder in the direction opposite to the pre-shear is less than that in the

standard forward direction. When lines are fitted through the data and those lines are

extrapolated they seem to converge at the y-axis. This is to be expected, since there

should be no difference when the pre-shear stress is zero, i.e., when there is no pre-shear.

The pre-shear incorporates a history in the powder and without this history there is no










difference between the two tests. This was also observed by Janssen et al. (2005) for

similar tests with a biaxial tester. When a least squares fit is performed with a common

intercept at the y-axis, this intercept is 0.71 kPa and the slopes of the linear fits change

from 0.26 and 0.11 to 0.28 and 0.09 respectively.

4
Standard U Reverse
fc, standard = 0.26 c1 + 0.79


3-




2-)





.. fc, reverse = 0.11 71 + 0.60



0
0 2 4 6 8 10
071 (kPa)

Figure 3-8: Flow function of BCR Limestone for a standard and reverse experiment,
measured with a Schulze cell, with 95% confidence bands for the linear fits
and extrapolation to or = 0.

Angle of Internal Friction

The angle of internal friction q usually varies with the applied stress. This means

that the yield loci (shown in Figure 2-2) are not exactly perpendicular, as shown in Figure

3-9. The behavior of the internal friction angle as a function of the major principal stress

appears to be different depending on the direction of shear during failure. In the case

where the pre-shear and failure are in the same direction the angle of internal friction

decreases from 38.50 at zero consolidation stress to an angle around 300 at higher







47


consolidation stresses. Conversely, in the case where the pre-shear and failure were in

opposite directions the angle of internal friction increases from 38.30 at zero

consolidation stress to about 450 at higher consolidation stresses. The difference in

behavior is explained by the difference in local stress patterns in the cell during failure.

50
e Standard Reverse 0,reverse = 0.65*o 1 + 38.3

45 -



40 -- -- -


35 -


30 -



25 standard = -0.88*or1 + 38.5


20
0 1 2 3 4 5 6 7 8 9 10
0 1 (kPa)

Figure 3-9: The angle of internal friction (i) as a function of the major principal stress for
a standard and reverse experiment, with 95% confidence bands for the linear
fits and extrapolation to 0r = 0.

Just like the unconfined yield strength, the angle of internal friction for the standard

and reverse experiments seems to converge at 0r is zero, i.e., when there is no history in

the powder. The value at which they converge is the same value as the effective angle of

internal friction. This shows that for very low stresses the yield locus coincides with the

EYL. Therefore, for these values it is very important that the real EYL is used so the

correct value for the effective angle of internal friction is found.









Normal Stress Distribution

The Tekscan technique is used to measure the normal stress distribution at the

bottom of the Schulze tester. For this, a Tekscan sensor is glued to the bottom of a

Schulze cell. A layer of sand is glued to the surface of the sensor to prevent slip of the

powder. Due to the sensor geometry only 21% of the cell bottom can be covered with the

sensor.

Figure 3-10 shows the normal force distribution of a section of the Schulze cell. It

can be seen that the force in the center of the angular cell is larger than at the sides near

the walls, which is to be expected (Bilgili et al., 2004).














Figure 3-10: Typical normal stress profile at the bottom of a slice of the Schulze cell.

The force is not constant in the angular direction of the cell. To show this, the

average force in the rectangular window in Figure 3-10 is recorded during a full rotation.

The results for three identical tests are plotted versus the angle of rotation in Figure 3-11.

It can be seen that the three experiments show the same trend. There are three to four

areas of high force. This would suggest that these areas mainly support the top lid and

load, since three or four points give a stable support.










1.8

1.6-

1.4 -

1.2 -




u- 0.8 -

0.6 -

0.4

0.2 -

0
0 90 180 270 360
Cell Rotation (degree)

Figure 3-11: Normal force on the bottom of a Schulze cell during steady state
deformation of BCR limestone.

Geometric Anisotropy

It is believed that this force distribution is the cause of a geometric anisotropy, i.e.,

due to tho tester geometry. This is explained by the following findings. Bilgili et al.

(2004) presented measurements of local stresses in the Jenike cell using Tekscan pads.

They found a skewed pressure profile in the Jenike cell with a higher stress toward the

leading edge of the cell. The same technique was used to measure anisotropy in the

Jenike cell. A failure in the same direction as pre-shear was compared with a failure in

the direction 180 degrees rotated from the pre-shear. This procedure is similar to that

used by Saraber et al. (1991) in their anisotropic Jenike shear measurements.

Figure 3-12 shows the pressure profiles measured at the cell bottom during the case

of pre-shear in the forward direction and failure in the forward direction. The pre-shear


orward direction. The pre-shear









profile shows the typical skewed profile observed by Bilgili et al. The failure profile

shows the same skewed behavior but at lower normal pressures. Figure 3-13 shows the

normal pressure measured at the bottom of the test cell for the conditions of pre-shear in

the forward direction and failure in the reverse direction. The same skewed profile exists

in the pre-shear profile and the failure profile, except that the failure profile is skewed in

the opposite direction. Because of this the local normal stress values during failure are

higher than the local normal stress during the pre-shear for approximately 1/3 of the

cross-sectional area. This suggests that a portion of the material in the test cell is

undergoing a transition of stress state during the failure mode of this test. The failure state

during reverse shear is a combination of actual failure conditions in a portion of the

material and attainment of a new lower steady state shear stress in the rest of the material.

This will obviously result in a lower overall shear stress value, which is measured for

anisotropic experiments. Saraber et al. suggested a renewed steady state to explain the

lower failure values for anisotropic shear. These new data indicate that Saraber et al. were

half right. The anisotropic behavior induced in the Jenike cell is due to the attainment of a

new critical state of stress, but occurs only in a portion of the cell.

A similar geometric anisotropy is expected in the Schulze shear cell. The results

show that there is an angular stress distribution which is similar to a skewed profile in the

Jenike cell. Since not the entire bottom of the Schulze cell is covered no comparison

between the steady state and the failure state can be made. Failure occurs instantaneously

and with only 21% of the cell bottom covered with the Tekscan pad, no conclusive

measurements can be made.










Steady State


shear direction fail direction


0.308


40 60 80
Distance Across Cell (mm)


Figure 3-13: Stress profile at the bottom of a Jenike cell during pre-shear and failure in
the forward direction, using silica powder.


Steady State


Failure


10.515
0.306


shear direction


fail direction


40 60 80
Distance Across Cell (mm)


Figure 3-13: Stress profile at the bottom of a Jenike cell during pre-shear in the forward
direction and failure in the backward direction, using silica powder.


Au2.5
a3

S2
" 1.5
CO)


0 0.5
0O


Failure












Modeling of Anisotropy in Biaxial Shear

This section describes work that has been done in collaboration with Dr.

Zhupanska. The anisotropy that is measured in a biaxial shear tester is compared with

DEM simulations. The experiments where performed with the Wall Flexible Biaxial

Tester (FWBT) (van der Kraan, 1996). This tester is an advanced powder flow tester that

allows for controlling stress and strain independently in two mutually perpendicular

directions. As a result, stress-strain curves for powder as well as unconfined yield stress

can be measured. The experimental study encompasses the comparison of standard and

anisotropic flow functions, similar to the tests as described above. It should be noted

though that no direct comparison is possible since the Schulze tester is a direct shear

tester and the FWBT is an indirect tester.

Biaxial Experiments

The experimental procedure of the tests as well as an extensive description of the

tester can be found in Janssen (2001) and Verwijs (2001). In short, the unconfined yield

stress is measured in the following way consisting of four steps. During the first step, the

powder is biaxially consolidated up to a chosen stress value. From this point, the powder

is pre-sheared (step two) by increasing the stress in one direction and decreasing it in the

perpendicular direction. During this step the mean stress is kept constant. At a certain

moment the powder is in steady state deformation and the stress cannot be increased

anymore. When this steady state deformation is reached, the third step is a relaxation of

the stresses to near zero. The final step is an increase of stress in the direction of major

stress during pre-shear while the stress in the other direction is kept zero. The maximum

stress that causes sample failure is the unconfined yield stress. A typical test result for the

standard experiment is shown in Figure 3-13.










In anisotropic experiments, the first three steps are the same but during the final

step the stress is increased in the direction of minor stress during pre-shear while the

stress in the direction that was major during pre-shear is kept at zero. So the stress profile

was rotated 90 degrees with respect to the standard experiment during step four. A typical

result of the anisotropic experiments is shown in Figure 3-15. It can be seen that

anisotropic failure takes more time and will result in a lower value.

10 -i o
[kPa]
8
failure
6-
1 2 3 4
4-

2-

0
0 5 10 time [h] 15 20

Figure 3-14: Typical result of a standard experiment with the Flexible Wall Biaxial
Tester, showing the stress in the x-direction (black) and y-direction (grey),
and the different steps of the test; biaxial consolidation (1), pre-shear (2),
relaxation (3), and failure (4).

16 C
[kPa]
12 -


8
anisotropic failure

4 1 2 3 4

0
0 5 10 time [h] 15 20

Figure 3-15: Typical result of an anisotropic experiment with the Flexible Wall Biaxial
Tester, showing the stress in the x-direction (black) and y-direction (grey),
and the different steps of the test; biaxial consolidation (1), pre-shear (2),
relaxation (3), and anisotropic failure (4).









Set-up Biaxial Simulation

The biaxial tests were simulated in 2D, omitting the z-direction, using the Itasca 2D

DEM code. Following the experimental procedure, a sample of size 0.075 x 0.075 m was

chosen. The simulations were done for about 7427 uniform circular discs. Disc size was

0.45 mm. A linear stiffness model was used for particle-particle and particle-wall

interactions with normal and tangential stiffness equal to 1.0*106 N/m for the particles

and 1.0*108 N/m for the walls.

The friction coefficient was 0.577 for particles and 1.0*10-5 for the walls. The low

friction coefficient for the walls was chosen since the FWBT is designed to have no wall

friction. Particle density was set to 2650 kg/m3 and the initial porosity was set at 0.16.

Strain control was set a wall velocity of 0.00001 m/s.

Following the real experiments in the biaxial tester described above, four different

stages were simulated with the DEM code (see Figure 3-16). The first step, biaxial

consolidation, was performed by uniform motion of all four walls towards the center of

the sample. The second step, pre-shear, was simulated by continuous motion of the x-

walls towards each other and motion of the y-walls away from each other. Just as in the

real experiment, at a certain moment, the sample started to deform with a steady state,

and stresses on the walls did not change anymore. At this point the third step, relaxation,

was invoked on the powder by releasing the walls until the stresses on the walls were

almost zero. As with the FWBT experiments, the fourth step was performed in two

different ways. For the standard experiment the x-walls were moved in the direction of

pre-shear until failure, while the y-walls did not move. For the anisotropic experiments

the y-walls were moved towards the center of the sample while the x-walls remained

stationary.
















biaxial consolidation pre-shear relaxation






normal failure anisotropic failure

Figure 3-16: The direction of wall movement at different stages of the biaxial test
simulation.

Stress and Porosity Distributions

During the different deformation steps the changes in stress and microstructure

were monitored. Figure 3-17 shows the normal stress in the x and y direction and the

porosity at the end of a biaxial consolidation to 17.2 kPa wall stress. It can be seen that

the stresses develop patterns symmetric about the x- and y-axis. The stress distributions

are not uniform as is generally assumed for biaxial consolidation. The magnitudes of the

stresses in the x and y direction are comparable, between 15 and 18 kPa. The porosity

distribution in the sample is symmetric with respect to the symmetrically applied load,

but there is a significant difference between the porosity in the center and at the walls.

1 18kPa r 18 kPa 0.2




I |l5kPa 15 kPa L. A I 1
y porosity

Figure 3-17: Normal stresses and porosity distributions after biaxial consolidation to
17.2 kPa.









The normal stress distributions and porosity after the second deformation step, the

pre-shear, are shown in Figure 3-18. It is apparent that the stresses in the x-direction are

twice as large as the stresses in the y-direction. This is to be expected since the sample is

loaded in the x-direction. There is a significant change in structure noticeable compared

with the biaxial consolidation. The porosity distribution is highly non-uniform in the

direction of compression but relatively uniform in the direction of relaxation. From this it

can be concluded that pre-shear causes significant structural anisotropy that will lead to

anisotropic behavior in subsequent steps.


126 kPa 12 kPa I 022



19|kPa 8kPa 0.12
or ay porosity

Figure 3-18: Normal stresses and porosity distributions after pre-shear.

After steady-state deformation, the walls were retracted until the stresses on the

walls were close to zero. A zero stress level on the walls is reached when practically all

contact forces are equal to zero at the particle contacts. Because of this, the stress

distributions as shown in Figure 3-18 disintegrated and became very homogeneous, i.e.,

all zero. The porosity increases slightly during relaxation but the pattern remains the

same as in Figure 3-18. Since the porosity distribution represents the sample structure,

relaxation of the sample cannot remove structural anisotropy introduced during the pre-

shear.

Figures 3-19 and 3-20 show the results of the fourth step, failure in the same

direction as pre-shear and perpendicular to that, respectively. As expected, a different

response was recorded during this step, depending on the direction of shear. When the










failure was performed in the direction of pre-shear the porosity distribution remained

practically the same, while the overall porosity slightly increased. In the case of failure

perpendicular to the direction during pre-shear a significant change in porosity

distribution can be seen. The overall porosity decreases slightly, while the distribution

becomes symmetrical, comparable to the distribution after consolidation. The stresses in

Figure 3-19, the standard experiment, are clearly higher than the stresses in Figure 3-20,

the anisotropic experiment. This is in agreement with the experiments performed with the

FWBT. It appears that there is a diagonal shear zone during failure of the standard

simulation, as can be seen most clearly from the oax plot in Figure 3-19. The anisotropic

simulation does not show such a clear zone. This seems to be in agreement with the

hypothesis proposed in the section above. During a standard experiment there is a real

failure, but during an anisotropic failure there is a change to a stress state that represents a

(partial) new steady state deformation. This is also confirmed with the porosity

distributions. The standard simulation does not show a change of structure, while the

anisotropic simulation shows a clear change of structure. It is expected that if the

anisotropic simulation is taken further in time, the anisotropy distribution becomes

similar to that during pre-shear with 90 degrees rotation. The porosity in Figure 3-19 does

not show a higher porosity for the diagonal shear zone, as would be expected. This could

be due to a limited resolution of the porosity calculation. It should be noted that the

biaxial experiments were stress controlled while the simulations were strain controlled.

The effect of stress and strain control will be discussed in chapter 3.









30 kPa 16 k 0.3



A 41 18 kPa 10 IlOkPa 0.1
oxx o7y porosity

Figure 3-19: Normal stresses and porosity distributions after forward failure.

S hh 11.5 kPa 27kPa 0.2




S 9.5 kPa 18 kPa 0.1
7xc 'yy porosity

Figure 3-20: Normal stresses and porosity distributions after anisotropic failure.

Microstructure

The distributions as shown above can also be studied at the particle or micro scale.

To describe the structure of the particles the concept of fabric measures generally

accepted in the field of soil mechanics can be adopted. This enables the investigation of

the effects of the microscopic structure on the macroscopic powder behavior, e.g., on the

development of anisotropy. To do this two fabric measures were calculated. One fabric

measure was related to the distribution of the contact normals and one was related to the

intensity distributions of the contact forces. In the 2D case these distributions can be

represented by an angular distribution that defines the fraction of the measure K(O) in a

certain direction 0 as shown in Figure 2-13. In practice the distribution is discretized in

sections zO. If the distribution function is circular there is no preferential direction of the

measure and the (overall) sample is isotropic for that measure.









The intensity of the distribution is determined by number in the case of contact

normals and by the total magnitude in the case of contact forces. In this study the overall

distribution for the sample has been studied. If the number of particles in a simulation is

large enough it is also possible to evaluate fabric measures for different positions in the

sample.

The changes in the fabric measures were monitored at the end of every step of the

simulation of the biaxial test. Slight changes were noticed in the orientation of contact

normals from a fully isotropic distribution after biaxial consolidation to a slightly

asymmetric distribution after relaxation. It is concluded that in the simulations no

significant changes in the orientation of contact normals could be observed. This is in

contrast to other studies (Oda et al., 1980; Rothenburg and Bathurst, 1989) where particle

contacts were preferentially aligned in the direction of the applied major principal stress.






consolidation pre-shear







normal failure anisotropic failure

Figure 3-21: Normal contact force intensity distribution after different steps of the
biaxial simulation.

The normal force distributions showed a clear difference between the different

simulation steps (see Figure 3-21). The distribution of the normal contact forces was

asymmetric and oriented in the direction of the major compressive stress after every step









except for the biaxial consolidation. There is no normal contact force distribution for the

relaxation step since all normal contact forces were near zero. Since friction was

practically absent on the walls of the cell, the tangential contact forces were two orders of

magnitude smaller than the normal contact forces and did not contribute significantly to

the anisotropic response. It should be noted that a simplified model is used for the

simulations, i.e., the simulation is 2D, the particle contact model is a simple linear elastic

model, the particles are all spherical and of equal size. Therefore, the results can not be

used quantitatively. It is clear though that qualitatively the simulations show anisotropy

that is similar to the anisotropy that is observed in the biaxial experiments.














CHAPTER 4
UNIAXIAL COMPACTION OF POWDERS

Uniaxial compaction or compression is a process that is largely used in industry to

make compacts like tablets or briquettes. There are roughly two operation regions when

considering compaction. There is a high stress region, used for tabletting, and a low stress

region, used for powder flow evaluation. The stresses used in the high stress region are

several magnitudes higher than the stresses encountered in the low stress region. This

chapter concerns with the low stress region, although some of the findings are applicable

in high stress compaction. The main difference is that the stresses encountered in the flow

regime are generally not high enough to permanently deform the particles, unless the

particles are very fragile. Also, the porosity in the flow regime is higher than in the high

stress compaction regime.

There are several uniaxial testers available commercially for the measurement of

compaction curves. For this research a new uniaxial tester has been developed. The

unique feature of the tester is its capability to deform a powder either stress or strain

controlled. It is generally accepted that stress and strain are directly and reversibly

related, making it irrelevant whether a tester is stress or strain controlled. This research

shows that this is not true for all powders. The compaction stress operating window of the

tester is zero to 35 kPa. The tester is built as a prototype for the development of a true

triaxial powder tester with independent stress and strain control in all principal directions.









Design of the Uniaxial Tester

The uniaxial tester consists of two main parts, the actuator system and the box that

contains the powder sample. The actuator system consists of two identical actuator units,

working in line and in opposite direction. The actuator system changes the sample holder

size to deform the powder. Figure 4-1 shows a picture of the uniaxial tester.













Figure 4-1: Picture of the Uniaxial Tester showing servomotor (A), linear stage (B), air-
cylinder (C), sample holder box (D), linear air bearing (E), guide bar (F), and
pressure regulator (G).

Each actuator unit consists of a linear stage for strain controlled motion and an air-

cylinder for stress controlled motion. The linear stage is a 100 mm wide 118 series

American Linear Motion (model PGA4-9-4/GB) position stage. It has 100 mm travel via

a 10 mm OD x 2 mm precision lead screw. The accuracy of the stage is 25 mn/mm. The

stage is driven by a Cool Muscle servomotor (model CM1-C-23L20) via a flexible

coupling. The motor has a high-resolution encoder of 50,000 units per rotation. The

motor has smooth motion even at low speeds.

The air-cylinders are custom made. They are drilled out of a rectangular aluminum

block and have a high accuracy 44.45 mm bore diameter. The pistons are made of white

Delrin. To get near frictionless motion the pistons are not sealed. The pistons are tapered

and the smallest clearance between the piston and the cylinder is about 0.05 mm, making









the cylinders leak. The cylinders are therefore called "leaking cylinders". The leaking

airflow lubricates the piston, making the piston near frictionless.

The leaking cylinders are pressurized with two independent pressure regulators

(Marsh Bellofram, type 3211). The range of the regulators is 0 350 kPa gauge,

corresponding to an actuator force of 0 543 N. The max flow of the regulators is 12

SCFM, which is enough to pressurize the leaking cylinders up to the maximum pressure.

The air-cylinders are mounted on the linear stages. The sample holder box is

connected with the two actuator units via shafts. A shaft includes a load cell, measuring

the applied load. The shaft can be locked for strain controlled experiments and unlocked

for stress controlled experiments. To stabilize and support the sample holder box there

are two guide bars per actuator unit. All guide bars are supported by two linear air

bearings. These air bearings provide a frictionless linear motion and are stiff enough to

support the guide bars and sample holder box. Figure 4-2 shows a top view picture of the

system showing the guide bars and the shafts.


















Figure 4-2: Top view picture of the Uniaxial Tester (without top lid) showing the air-
cylinder (A), shaft (B), linear air bearing (C), guide bar (D), and sample
holder box (E).









The sample holder box is an important part of the tester. Due to friction at the

walls, the geometry of the sample holder influences the test results. The design of the

uniaxial tester with a separate actuator system and sample holder allows the use of

different box designs. The different box designs that are built are rectangular in shape.

The box design can also be cylindrical but that would make the filling process

complicated. Figure 4-3 shows the different rectangular box designs.


Two Half Boxes



Two U-Boxes




Channel




Figure 4-3: Different sample holder box designs.

The difference between the boxes is the direction of the shear stress at the side,

bottom, and top walls. In the "two half boxes" design (also shown in Figure 4-2 without

the top), the shear stress at the bottom and one of the side walls is in the same direction,

while at the top and the other side wall it is in the opposite direction. In the "two U-

boxes" design the shear stress is in the same directions at the top and bottom wall, and in

the opposite direction at the sidewalls. In the "channel" design the shear stresses are in

the same direction at all walls when only one piston is moving. The stress directions

change approximately midway of the bottom, top and side walls when both pistons are

moving.









The position of the box parts is monitored with two linear potentiometers (Omega

model LP804). The potentiometers are mounted on the outside of the air-cylinders and

attached to the back of the box. This enables monitoring of the position of the cylinder

piston and calculation of the sample size. The two loading walls both have a constant

cross sectional area of 127 x 127 mm2. This gives a ratio of the pressure in the leaking

cylinder over the stress on the powder of 10.4.

Uniaxial Experiments

Initial tests with the uniaxial tester showed that the two half-boxes sample holder

design could not be used at high stresses. When a powder is uniaxially compacted, there

is a resulting normal stress at the side walls. In the two half-boxes design, the normal

stress on the side walls causes a moment on the air bearing supporting the sample holder.

At a compacting stress above about 8 kPa the moment on the bearings becomes larger

than their rating. This causes friction in the bearings, which can damage them. It also

causes hysteresis in loading-unloading loops, affecting the measurements. In the two u-

boxes design, the moment of opposite walls is approximately offset, avoiding the

problem.

The experiments reported in this dissertation are performed with the two u-boxes

sample holder. The tester is filled by scooping powder through a sieve into the sample

holder box. The sieve breaks up any agglomerates and ensures a more or less consistent

packing from sample to sample. The excess powder is scraped off from the top and the

top lid is placed.

Two types of experiments are performed; stress controlled and strain controlled.

Stress controlled means that the tester dictates the stress path and the resultant powder

deformation is measured. For the experiments reported the stress path is set at a constant









stress increase on the powder. Strain controlled means that the tester dictates the

deformation path and the stress response of the powder is measured. For the experiments

shown the strain path is set at a constant strain velocity. For both types of experiment,

one actuator unit is locked, resulting in uniaxial consolidation with one moving wall.

The two powders that are tested are Microcrystalline Cellulose powder (PH101)

and Polystyrene powder. The mean particle size of Microcrystalline Cellulose is 50 jnm.

The mean particle size of the Polystyrene powder is 40 ,um with a narrow size

distribution (see appendix B).

Uniaxial Compaction of Microcrystalline Cellulose

Microcrystalline cellulose is compacted stress controlled at a rate of 0.01 kPa/s and

strain controlled at a rate of 0.48 mm/min. The resulting compression curves are shown in

Figure 4-4. It can be seen that the two curves do not differ significantly. That is what is

mostly reported in literature. That is why it is generally assumed that stress control and

strain control produce states that are indistinguishable.

For operation of process equipment it does matter though whether stress control or

strain control is used to run a process. This is due to the shape of the compaction curve. It

can be seen that the stress increases exponentially. So for high stresses the stress can

double with a minimum deformation. In a tablet press the stamp and die are strain

controlled and set at a certain deformation. A minimal deviation in powder load will

therefore result in a significant compaction stress difference. The compaction stress

defines the strength of the tablet. If for quality purposes the tablets need to have a

controlled strength, e.g., pharmaceutical pills, the press should be stress controlled. If the










size of the tablets is the foremost concern, e.g., precision cast parts, the press should be

strain controlled.

20
-stress control
-strain control
16 -



12 -


8 -



4-



0
0 0.04 0.08 0.12 0.16
Strain (-)

Figure 4-4: Uniaxial compaction of Microcrystalline Cellulose PH101, stress controlled
(grey) and strain controlled (black).

Uniaxial Compaction of Polystyrene Powder

Polystyrene is a powder that exhibits stick-slip. The powder is stress controlled at a

rate of 0.045 kPa/s and strain controlled at a rate of 0.2 mm/s and the results are shown in

Figure 4-5 and 4-6, respectively. The difference between Figure 4-4 and Figures 4-5 and

4-6 is apparent. The stress-strain curve for microcrystalline cellulose is smooth while the

stress-strain curves for polystyrene are very jagged, showing stick-slip.

There is a clear difference between the stress controlled and strain controlled stress-

strain curves of polystyrene. In the stress controlled experiments the slips exhibit

themselves by quick deformations, plotted as near horizontal lines in the curve.






































0.02 0.04 0.06 0.08 0.1 0.12
Strain (-)


5



0
0



Figure 4-5:


30 --



25



20






10



5



0
j-


Stress controlled uniaxial
polystyrene powder.


compaction and relaxation curve of 40 atm


0 0.02 0.04 0.06 0.08 0.1
Strain (-)


Figure 4-6: Strain controlled uniaxial compaction curve of 40 atm polystyrene powder.


0.14









The sticks on the other hand endure a long time and form near vertical lines in the

curve. In the stress controlled experiment there are few, but very violent stick-slip events.

Due to the fast slip in which the powder deforms up to a few millimeters in a fraction of a

second, powder is blown out of the sample holder. The air that is present in the powder

voids has to escape when the sample deforms. The tester walls are solid, so the air has to

escape through the narrow gaps between the walls. When this happens very fast, the

velocity of the air through the gaps becomes high and the air will entrain some of the

powder.

In the strain controlled experiment a slip exhibits itself by a fast stress decrease,

shown as a vertical line in the curve. During the stick phase the stress increases until the

next slip. The stick-slip events are very frequent but due to the controlled deformation

rate the events are not as violent as the stress controlled slip events. The stick-slip signal

is investigated in detail in chapter 5.

The experiments with polystyrene show that stress control and strain control are not

reversible for certain powders. Starch powder showed the same behavior as polystyrene.

The conclusion of the work is that powders that show stick-slip will behave differently

when stress controlled or strain controlled, making the two control systems non-identical.

Modeling of Uniaxial Compaction

Using the Itasca PFC 2D Discrete Element Method the uniaxial compaction is

investigated. The focus was on the influence of the sample holder geometry and the

influence of stress and strain control on the powder structure. For the simulations a linear

contact stiffness model for circular particles is used. The 10,400 simulated particles range

in size from 80 |tm to 1.2 mm and have a uniform distribution. The friction coefficient is









taken to be 0.577 between the particles and 0.364 between the particles and walls. These

values are estimates but it will be shown in chapter 6 that the value of the friction

coefficient should be measured accurately since it is of major importance. The cell size

for the simulations is 63.5 mm x 63.5 mm and the initial 2D porosity is 0.17.

Influence of Cell Geometry

As discussed at the beginning of this chapter, the direction of the shear stress on the

walls will be different for different sample holder geometries. The stress distribution and

structure of the particle ensemble in the two half boxes and the two u-boxes design of the

uniaxial tester are investigated. The two half-boxes design is called geometry 1 and the

two u-boxes design is called geometry 2 and they are shown in Figure 4-6. It should be

noted that geometry 2 is a cross-section of the center of the u-boxes design and represents

more correctly the 2D simulation of a channel design with one moving wall.












Figure 4.6: Sample box geometries used for the simulations; geometry 1 (left) simulates
the two half-boxes design, geometry 2 (right) simulates the two u-boxes or
channel design.

The initial particle ensemble is the same for both simulations. During the

simulation of geometry 1 the left and bottom wall move to the right while the other two

walls are stagnant. During the simulation of geometry 2 only the left wall moves to the

right while the three other walls are stagnant. The geometries are both strain controlled

deformed with a strain rate of 9*10-11 step-1. The time scale is given in steps and not in










real time since it reduces the computation time. In Figure 4-7 the stresses on the x- and y-

walls is given for the two geometries as a function of the time steps. In this case the x-

axis is equivalent to the strain. It can be seen that there is no significant difference in the

overall wall stresses for the two geometries.

600
Geometry 1, x-walls
-Geometry 1, y-walls
500 -Geometry 2, x-walls
-Geometry 2, y-walls

400 -


1 300 -


200 -


100 -


0
0.OOE+00 8.00E+06 1.60E+07 2.40E+07

Time (Step)

Figure 4-7: Overall stress on the x- and y-walls of two cell geometries during uniaxial
compaction.

The stress distributions in the particle ensembles do show a difference between the

two geometries. The difference of the normal stress distribution in the y-direction is

minimal but the normal stress distribution in the x-direction is significant. This is shown

in Figure 4-8. Geometry 1 shows high stress region diagonally from the top left to the

bottom right while geometry 2 shows a more symmetric stress distribution with a high

stress band in the middle from left to right with high stress points in the left comers. This









could indicate that there is a shear plane in geometry 1, something that is generally

considered not possible in a uniaxial tester.





400





350




Figure 4-8: Distributions of normal stress in the x-direction for geometry 1 (left) and
geometry 2 (right). The unit of the scale is kPa but for a 2D simulation.

In Figure 4-9 the fabric tensors are shown for the two simulations. It can be seen

that the angular distribution of the contact normals is isotropic. This means that there is

no preferential contact direction, similar to the results in the previous chapter. There is a

clear anisotropic distribution of the magnitude of the normal forces. The difference in

anisotropy between the two different geometries is not significant though. The magnitude

is slightly larger for geometry two, but this is because this simulation is stopped at a

slightly higher stress level. The difference between the two simulations is apparent in the

angular distribution of the tangential contact forces. Surprisingly, geometry 1 is more

isotropic than geometry 2. It should be noted that the magnitude of the tangential forces is

fifty times smaller than the magnitude of the normal forces. Therefore, the influence on

the overall bulk behavior will be less significant. This would exclude the possibility of

shear zones during powder compaction.










750 -





-750


Number of Contacts


-750
Number of Contacts


230





-23-



-230


Magnitude Normal Contact Force (N)


Magnitude Normal Contact Force (N)


4


Magnitude Tangential Contact Force (N)


Magnitude Tangential Contact Force (N)


Figure 4-9: Fabric tensors for uniaxial compaction using two different geometries. The
tensors at the left are for geometry 1 and the tensors at the right are for
geometry 2. Shown are the angular distributions of the particle contact
normals (top), normal force magnitude (center), and tangential force
magnitude (bottom).


750


30

~7 )


)









It should be noted that the fabric tensors in this work are used as overall

parameters, covering the whole sample. This means that there could be a narrow shear

zone with a high shear stress that is not captured because the volume of the shear zone (or

area in 2D) is much smaller than the overall volume. Therefore to reach conclusive

evidence of shear zones during powder compaction it is necessary to use local fabric

tensors to identify the structure in different regions in the powder to investigate possible

shear regions. For that a larger number of particles is necessary to make a statistically

correct angular distribution.

Pseudo-Stress Control versus Strain Control

The experiments with the uniaxial tester showed that for certain powders stress and

strain control give distinctively different results. Some testers and process equipment are

said to be stress controlled while they actually are pseudo stress controlled. Pseudo stress

control is indirect control. The stress (or usually load) is measured with a sensor. Through

a feedback control the stress on the powder is maintained by a strain device, usually a

motor with a lead screw. If the stress is too low, the motor moves the walls inwards, and

if the stress is too high, the motor moves the walls outwards. This is called servo control.

Measurements with pseudo stress control can be seriously biased. The feedback

control needs to be much faster than the processes in the powder. Since we do not know

the fundamental powder behavior, we do not know for sure that our feedback is fast

enough. It is very well possible that tester artifacts are measured instead of powder

properties. It is not very likely that a pseudo stress controlled tester can measure stick-slip

correctly. The slips in the powder are too rapid.

To investigate the influence of pseudo stress control, uniaxial simulations are

performed using the servo control method. Since the Itasca code does not provide a real










stress controlled mechanism the servo results are compared with pure strain control. The

simulations are performed using geometry 1. The two gain factors that are investigated

are 0.001 and 0.005. A larger gain is faster, enabling a quick response, but it is also

coarser. When controlling a powder, the large gain can cause significant overshoots.

The results are shown in Figure 4-10. As expected, the larger gain results in a larger

initial control step. The stress quickly increases to about 100 Pa for a gain of 0.005. After

some strain the two pseudo stress controlled curves seem to converge. This would mean

that only for low stresses pseudo stress control biases the results. This initial difference

might not be large enough to change the particle structure to cause different behavior. It

is likely that it will largely influence the behavior for a powder showing stick-slip. The

difference in stress-strain curves is a due to a combination of tester characteristics and

powder characteristics.

600
-pseudo stress control (0.005)
-pseudo stress control (0.001)
500 -strain control


400 -


w 300 -


200 -


100


0
0 0.0005 0.001 0.0015 0.002 0.0025
Strain (-)

Figure 4-10: Consolidation curves for uniaxial simulations; strain controlled (red) and
pseudo stress controlled with a gain factor of 0.005 (blue) and 0.001 (black).















CHAPTER 5
STICK-SLIP IN POWDER FLOW

Some powders exhibit stick-slip during deformation but most do not. A very well

known powder that exhibits stick-slip is snow. When we compress snow to make a

snowball or snowman, but even more when we walk on snow, we can feel and hear the

stick-slip. The snow does not deform in a smooth manner but in many short steps,

quickly following each other. These steps cause the crunching sound that we hear. In this

research snow was not chosen as a model powder. Besides expensive climate control,

snow particles or flakes are fractals, which makes the modeling of the system very

complex.

In the kitchen we can find two powders, flour and starch, that initially seem very

similar. Starch is slightly whiter than flour, which enables us to see the difference. When

we feel the powders they appear very different though. Starch exhibits stick-slip, which

we can feel and hear when we squeeze it between our fingers. Flour does not show this

phenomenon.

This chapter investigates the macroscopic response of stick-slip in powders. Both

the magnitude and frequency of stick-slip are investigated as a function of both operating

parameters and powder properties. The two operating parameters that are varied are the

normal force that is exerted on the powder and the deformation rate or shear velocity. The

powder parameters that are varied are the particle size distribution and the moisture

content.









Introduction to Stick-Slip

In the field of tribology much research has been conducted on stick-slip between

surfaces. Stick-slip is a non-smooth motion, in jerks, and can be regular (periodic) or

irregular (erratic or intermittent). It is generally considered that the difference between

static friction and kinetic friction causes the material to move intermittently (Bowden and

Tabor, 1950). The difference between static and kinetic friction is due to the surface

roughness. The microscopic contact area between surfaces will be larger in static

conditions than during sliding, resulting in a higher static friction.

Generally, lubrication reduces or eliminates stick-slip but even lubricated surfaces

can exhibit stick-slip. Thompson and Robbins (1990) did molecular dynamics

simulations of a lubrication layer between two plates. The bottom plate was kept

stationary while the top plate is moved attaching it with a spring to a translating stage

with a constant velocity. They attributed the stick-slip in the boundary lubrication to a

transition from a crystalline state of the lubricant to a fluid state. A thin layer of lubricant

can form a structure that is ordered like a crystal, which periodically melts due to the

shear between the plates, causing the slip, and recrystallizes, causing the stick. The

results indicate that stick-slip in boundary layers is caused by thermodynamic instabilities

during the sliding phase rather than a dynamic instability. It is observed that stick-slip is a

function of the shear velocity. The frequency of stick-slip increases generally with the

velocity.

Berman et al. (1996) present three different stick-slip mechanisms. The first is the

surface topology or roughness model as depicted in Figure 5-1. Due to the asperities on

the contacting surfaces, the friction force increases when an asperity on one surface has

to travel over an asperity on the opposite surface. When the two asperities have passed









each other, the friction force will quickly drop due to the quick fall, e.g., from B to C.

Stick-slip due to surface roughness is irregular, unless the surface roughness has a regular

pattern. The stick-slip frequency increases with the shear velocity while the amplitude

decreases.




A D.
0E
B Time
D





Figure 5-1: Schematic representation of stick-slip due to surface roughness.

The second stick-slip mechanism is called distance-dependent model or creep

model. During shearing, due to the adhesion between asperities, the surfaces have to

creep a distinct distance before the adhesion breaks and the surfaces slip. This type of

stick-slip is mainly observed for dry systems and results in a regular, periodic, signal. It is

often used to model rock-on-rock sliding in geology. The amplitude decreases with

increasing shear velocity.

The third stick-slip mechanism is the velocity-dependent model. This model applies

to lubricated systems. At low velocities stick-slip is present but it can vanish above a

certain critical velocity. From their simulations, Robbins and Thompson (1991) found

that for lubricated surfaces this critical velocity is equal to the velocity of the top plate

just before recrystallization of the lubrication boundary. In their system the critical

velocity is a function of the mass of the top plate and not an intrinsic lubricant property.

Yoshizawa and Israelachvili (1993) performed similar simulations for different









lubrication species. They found a critical velocity dependent on the spring constant of the

spring pulling the top plate. So the critical stick-slip velocity between two surfaces is a

combination of the surface and lubrication properties as well as the test system.

The conventional models generally cannot explain stick-slip in granular material.

Many phenomena are similar for both systems though. Friction in granular material can

be regular and irregular. A critical velocity is found for both friction systems. The

dependency of the friction force on the velocity is often represented as a bifurcation as

depicted in Figure 5-2 (Bucklin et al., 1996; Kolb et al., 1999; Lacombe et al., 2000)









TmV, Vc
Time Shear Velocity

Figure 5-2: Schematic representation of the bifurcation of the friction force as a function
of the shear velocity. The bifurcation point is the critical stick-slip velocity vc.

Budny (1979) published what might be the first publication about stick-slip in

powders. He used stick-slip as an alternative method to characterize powder flow. His

experimental set-up comprised of an Instrom pulling an aluminum plate over a table. The

table and aluminum plate were covered with sandpaper to provide friction. In between the

plates a thin bed of powder was placed. The model Budny proposed to describe stick-slip

is a modified Mohr-Coulomb model, shown in equation 5-1. The model can be used to

find the internal frictional coefficient /u, as well as a stick-slip friction coefficient /U, as a

function of the normal force F, and the stick-slip frequency co. The term /F,n is called the

stick-slip friction and is the difference between the maximum and minimum friction.










Ffncon =, ,Fn + ,uF, sin ot 5-1

Budny found a decrease in the stick-slip friction, or magnitude of the stick-slip,

with increasing sled velocity. This is consistent with Figure 5-2 but he did not report a

critical velocity for a range up to 100 inches per second. He reported a decrease in stick-

slip magnitude with increasing particle size.

Bucklin et al. (1996) used a similar setup to measure stick-slip in a granular wheat

bed. They pulled a plate of wall material through a bed of powder, thus both sides of the

plate were in contact with the grains while the grain bed was loaded with weights.

Bucklin et al. reported a critical stick-slip velocity and a bifurcation similar to Figure 2-5.

The lower bound (minimum friction) seemed to be a continuation of the curve without

stick-slip, while the upper bound (maximum friction) seemed to divert from the

bifurcation point upward. No dependency on the pressure was observed for the critical

velocity.

Albert et al. (2001) measured the force on a rod in a rotating granular bed and

reported stick-slip. The rod was connected to a load cell with a spring. The spring

constant could be changed by exchanging springs. Below a certain spring constant no

dependency on the spring constant or the shear velocity could be found for the friction

force. The product of the spring constant, the velocity and the inverse of the frequency

was constant.

Depending on the depth of the rod in the granular bed and the particle size, the

stick-slip signal was either periodic, random, or stepped (see Figure 5-3). Albert et al.

concluded that the stick-slip behavior is due to jamming of the rod in the granular bed.

The development of force chains in the bed is dependent on the particle size as well as









the depth of the rod. A linear increase of the magnitude of stick-slip with rod diameter

was reported.















Time (s)

Figure 5-3: Different types of stick-slip according to Albert et al. (2001).

Schulze (2003) used the a ring shear tester to investigate the time and velocity

dependency of stick-slip. He showed that the shear stress of powders can either increase

or decrease with increasing shear velocity, depending on the powder. When a powder is

sheared, stopped for a while, and sheared again, either an increase or decrease in shear

strength can be measured, again depending on the powder. Schulze accredits this to the

creep behavior at the particle contacts. A powder that shows an increase in shear stress

after a stop period will show an increase in stick-slip magnitude with decreasing shear

velocity. A decreasing shear velocity means that the stick time is longer and thus the

creep can work longer, creating a higher shear stress.

Volfson et al. (2004) used molecular dynamics to simulate stick-slip in a Couette

cell and compared it with a theoretical model. Although they call it granular, the

simulations represent a thin lubrication layer rather than bulk solids.









Experimental Setup

In this chapter two types of testers are used, the Schulze cell and the Uniaxial cell.

In Chapter 2 the Schulze cell is described in detail. The signal from the Schulze load cells

was recorded digitally using a Newport INFCS-001B amplifier and a LabJack U12 data

acquisition device. The experimental setup for the Uniaxial tests is described in chapter 4.

In this chapter the stick-slip data that are presented in chapter 4 are investigated in depth.

Two powders that show stick-slip are used in this research. For its wide use in

industry and easy availability, cornstarch has been chosen. The starch particles are

somewhat spherical with flattened sections as can be seen in Figure 5-4. The mean

particle size is 13 |tm with a standard deviation of 7 |tm. The second powder that is

chosen is a polystyrene powder. The powder is produced by Kodac Company and is used

as a milling medium for pigment milling. The polystyrene powder is available in several

particle size distributions. The two different sizes used have a mean particle size of 9 |tm

and 40 |tm with a standard deviation of 6 and 9 |tm, respectively. As can be seen in

Figure 5-4 the polystyrene particles are very spherical. That makes the powder a good

model powder for DEM simulations.


Figure 5-4: SEM images of cornstarch (left) and polystyrene particles (right).









In the Schulze tester the angular velocity of the base can be adjusted between zero

and 0.025 RPM. The linear shear velocity in the angular cell is not constant throughout

the sample. At the outside of the sample the linear velocity is twice as large as at the

inside. To calculate a representative (overall) velocity, the weighted velocity v is reported

as calculated with equation 5-2. In this equation R, and Ro are the inner and outer

diameter of the sample and o) is the angular velocity.

4 (R3 R3
V = -T 0 0 5-2
3 R2 R2
0 1 )

For the experiments where the moisture content of the powder was varied, the

moisture content was determined using an Ohaus MB 45 Moisture Analyzer. To reach a

required moisture level the moisture content was measured and an additional amount of

water calculated and added. The powder was mixed for 20 minutes using a conventional

kitchen blender. The required moisture level was checked using the moisture analyzer.

Results of Schulze Tests

Figure 5-5 shows a typical stick-slip signal from a Schulze Shear Cell. During the

rise of the shear stress, the stick phase, the powder does not shear. When it reaches a

maximum stress, it slips or shears quickly until a minimum stress value is reached. The

stick period is long compared to the slip period. It can be seen in the figure that, after a

short initial period, both the periodicity and the maxima and minima are constant for a

certain experiment. Therefore the signal can be called a periodic steady state. The

maxima correspond to what is traditionally considered the steady state value.

To verify that there is no powder deformation during a stick period, a linear

displacement potentiometer was attached to the top lid to measure the lid displacement

(see Figure 5-6). The Omega LP804 potentiometer is attached to the lid through a slot






84


which enables the lid to move up or down with the powder. The potentiometer has a

linearity of 1% (full scale), a repeatability of 12 |tm, a hysteresis of 25 |tm, and an

incremental sensitivity of 0.13 |tm.

1.5 I


vv VVVVVVVV
,, / m


05 stick
c 0.5

slip

0
8 9 10 11 12
Time (s)

Figure 5-5: Typical stick-slip signal from a Schulze cell.


Figure 5-6: Picture of the potentiometer fixture to the Schulze cell lid.

During an experiment the base, which contains the powder sample, rotates with a

constant angular velocity. If the sample does not deform during the stick phase the top lid

will move with the powder and there will be a deformation of the most flexible parts in









the system. Since the stick-slip events have a constant maximum and minimum shear

stress and period, this deformation is an elastic deformation. Appendix C shows the

stiffness calculation for the system of base, powder, top lid, tension bars, and load cells.

The base of the tester and the top lid are very rigid. The load cells and tension bars are the

most flexible but account for only 50% of the elastic deformation. The rest of the elastic

deformation is within the powder. Therefore, if the displacement is measured accurately,

stick-slip can be used to measure the bulk elastic modulus for a powder.

Figure 5-7 shows the horizontal top lid displacement (at the potentiometer point)

together with the shear stress. Since the horizontal top lid displacement is measured

relative to the tester table, the displacement oscillates between a maximum and minimum

value. The absolute value of these displacements is of no significance, only the relative.

The vertical lines indicate the slips of the powder, while the jagged curved sections

indicate the stick phases. It can be seen that the stress drops as soon as the top lid slips. In

Figure 5-8 the horizontal displacement data are corrected for the rotation of the base,

making the base with the powder sample the frame of reference. It can be seen that the

top lid has a clear stepped movement with long periods of stick, the horizontal sections,

and very short slips, the vertical sections. A slight deviation from the horizontal plane is

noticeable immediately after the slip and just before the next slip, indicated in the Figures

with an A and C, respectively. This would mean that there is relative motion between the

top lid and the base with sample during the stick phase. At the end of the stick phase

(section C) this is due to creep of the powder. It can be seen in Figure 5-7 that the slope

of the deformation changes, due to the dependency of creep on the increasing stress. The

movement immediately after the slip is a shear at the rate of the tester. The horizontal