Chemical EngineeringSchool of Process, Environmental and Materials Engineering

MSc Chemical Engineering

CENG 5120 Dissertation Project

Electro-Mechanical Modelling of Charged Particulate Systems: Macroscopic Properties and Microscopic Origins

Student: Mohammad Ashar Sultan Supervisor: Dr. S. J. Anthony

September 2005

I

CONTENTSContents Acknowledgements Summary I III IV

Chapter 1 Introduction 1.1 Introduction to Electrostatic Force1.2 Objectives 1 2

Chapter 2 Literature Review Chapter 3 Theoretical Background3.1 Force Laws 3.2 van der Waals Forces 3.3 Electrostatic Forces 3.4 Electrophotography 3.5 Typical Application Involving Electrostatic Charging Of Particles 3.5.1 Application Of Piezoelectric Effect In Solid State Battery 3.6 Inter-Particle Force Equations 3.7 Force Model Commonly Used For Toner Particle Systems

4 88 9 11 11 14 14 15 25

Chapter 4 Computer Experiment4.1 Particle Simulation Software 4.2 Role Of The Computer Experiment 4.2.1 Setting Up Computer Experiments 4.3 Types Of Modelling 4.4 Discretisation 4.4.1 Discrete Particle Modelling 4.4.2 Discrete Particle Simulation 4.5 Typical Force Separation Relation Of Toner Particle (Non-Contact Regime) 4.6 Simulation Experiment 4.6.1 Simulation Types

2929 29 30 31 32 32 33 35 35 38

Chapter 5 Results And Discussions

41

II

Chapter 6 Conclusions And RecommendationsList Of Captions Nomenclature References

5456 57 58

III

AcknowledgementsI would like to express my deepest gratitude to Dr. S. Joseph Antony, University of Leeds, enthusiastically and patiently guided me during the course of my project. Dr. S. Joseph Antony always shared his valuable experience not only as researcher but also as a practitioner. His profound knowledge significantly aid me in implementation and development of this dissertation. I would like to thank my parents for all their support on every step of my life. I would like to dedicate this dissertation project to my parents for all the sacrifices they went through in order to make possible the excellent experience of being part of the research with the Department of Chemical Engineering at University of Leeds.

IV

SUMMARYElectrostatic force on particles can arise in the presence of net electric charge on the particles or externally applied electrical field. This is of immense commercial importance in several engineering applications, for example, the field of xerographic, photographic, semiconductor industries, pharmaceutical, biotechnology and energy sectors are a few to mention. Particles can acquire charge in various ways such as by contacting with other materials, by corona ions, by induction in an externally applied electric field. The influence of electrostatic force on the bulk behaviour of particles is still a subject of debate and a virgin research area. In this thesis work, an attempt has been made to investigate the role of electrostatic force, generated by charging them, on the macroscopic and several micromechanical features on a fundamental level. The study is based on advanced discrete element simulations, in which individual properties of the particles and the electrostatics force acting between them can be specified. Newtonian Law. The dynamics of the particles are a result of the net electrostatic forces acting between the neighbouring particles, governed by The results show that the bulk behaviour of the particles is significantly influenced by applying electrostatic charging. Detailed analysis is performed to investigate the role of variations in the short-range electrostatic force, pull-off forces due to (Johnson, Kendall and Roberts) JKR and Derjaguin, Muller and Toporov) DMT theories and the mechanical contact forces (due to particles touching/overlap). For the first time, attempts have been made to study the extent to which the contact forces acting between particles re-align due to charging and its subsequent consequences on the bulk strength characteristics of the granular assemblies. We find that, among all the individual force components, the variations in the short-range electrostatic forces dominantly affect the bulk properties of the granular assemblies and this is often ignored in the limited existing studies. The fundamental understanding gained from this study would help towards achieving more efficiency in processing industries that handle their products in a powder form.

V

CHAPTER1: INTRODUCTION1.1 Introduction to Electrostatic forceParticle interactions at a microscopic level are less understood with regard to the strength characteristics of charged particulate system. Multiple particle forces exist within this type of systems including electrical and magnetic fields, air drag, friction, gravitational forces, attractive/repulsive forces, contact charging/electrification, triboelectrification etc. Electrostatics has found its basis in many major industries related to mineral, communications and other major engineering fields. The application of electrostatics is being used in several industrial applications, for example, in agriculture, paints, aeronautical and space, pharmaceuticals. In most of these industries, electrostatic force properties of the particles are used to collect, separate, deposit lightweight or heavy particles [36]. The application of electrostatics is based on inspired used of electrostatics force such as Coulomb force, image force, gradient force etc. The Coulomb force is most widely used and strongest force present. The magnitude of charge and coulombic force is proportional to the surface area of an object. As a result the effect of coulombic force relative to the mass-force as the major counter force becomes approximately proportional to its specific surfaces. So the particles, fibres and sheets all have a large specific surface are the major objects of application where the electrostatic force becomes dominant over the mass-force to control the motion. The direction of the forces can be changed by merely changing the polarity of the electrode [40]. Many advantages can be taken from exploiting the characteristics of the electrostatics force. Firstly its ability to control the trajectory of the particles in the size range from microns to millimetre. Secondly, is the dependence of electric fields upon the inverse square law that results in rapidly increasing forces as the separation distance reduce and surface comes closer [12]. Computational simulations using various modelling techniques continuously evolve to become more accurate and efficient. Methods and detailed physical models are applied to qualitatively and quantitatively describe and characterize the behaviour of real particle ensembles in many simulation environments that are at the current edge of research. In a fiercely competitive market, computational ability and simulation research is imperative to understanding of problems and improvement of processes.

1

The uses of different computational simulation methods to apply models describing inter-particle and particleobject interaction within electrical fields have been explored for some time. However problems occur in this area due to the high degree of computational time and expense required. Difficulties are caused in the calculation of long-range coulombic interaction in many particles ensembles. The introduction of mechanical forces in addition to electrical fields further complicates particle interaction and their bulk behaviour. It is vital that models used within computer simulations should be chosen carefully to most accurately represent a given system.

1.2 ObjectivesIn this thesis at the first instance, we review the literature on the different types of inter-particle force models available. Then for a chosen test particle system (toner particles), we account appropriate inter-particle forces acting on a toner particle system. In addition to the electrostatic forces, the mechanical forces acting between the particles are also accounted. The electrostatic forces are bifurcated into the short-range and long-range contributions. Extensive simulations are performed to I. Investigate the extent to which the short-range and long-range forces contribute towards the bulk behaviour of the assemblies and II. Study the role of single particle properties, particularly, the impact of interparticle friction upon the macroscopic phenomena that are operative in charged particulate systems. To the best of our knowledge, the study is entirely new and provides several key understanding on the features of charged particulate systems. In this study, a discrete particle dynamic simulation and analysis are performed to investigate the micro-macroscopic characteristics of charged granular assemblies. Objective 1 To find out different inter-particle force equation in the presence of electric field, i.e. to find out the effect of electric field on the particles in a granular assemblies. Objective 2 To select an appropriate inter-particle force model equation for toner particles, between electrostatic force and separation distance by solving appropriate charge potential equations.

2

Objective 3 Perform simulations to probe the role of inter-particle forces model on the bulk behaviour of charged toner particulate systems. All of the objectives involve the creation and adaptation of Fortran program to generate code that represent parameters to represent a system environment in its geometry, materials, physics applied and runtime loops.

3

Chapter 2: LITERATURE REVIEWParticle modelling with regard to electrostatics became viable and attractive to commercial companies as presented by Hockney (1963) in his 2D fast direct solver for Poissons equation [30]. This solver included field interpolation methods that led to greater computational speed and economy. This solver allowed a reasonable ability to simulate a number of particles (around 2000) within a 2D system. Harlow (1964) [24] developed the higher-order interpolation scheme, the particle in cell PIC technique that is later utilised extensively by the Berkley group (1969) [18] in fluid dynamics simulation. With the development of faster interpolation schemes [35], fast fourier transforms etc and faster computers the Particle Mesh (PM) method was developed by Birdsall and Langdon [10]. Their research was related to simulation of hot gas plasmas (with relation to plasma physics) and is documented by Birdsall and Langdon [10]. The concept of introduction of direct particle-particle interactions in addition to the particle-mesh modelling technique proved fundamental to Hockneys works as he later developed a two-dimensional computational model that utilises the hybrid particle-particle particle-mesh technique [30]. Shaw and Retzlaff (2003) [59, 60, 62, 63] furthered Hockneys work within a system that is able to facilitate a three dimensional particle in cell hybrid model like system that uses a hybrid algorithm [30]. They use an object orientated programming approach to Hockneys work within a programming set. Shaw and Retzlaff (2003) approach also offers the ability for force models to be applied as boundary conditions within simulation. This was done in conjunction with developing more detailed short range models used within simulation, with use of, distinctive particle-particle and particle boundary models and adhesion /cohesion parameters providing empirical relationship taken from electrostatic detachment cell experimental testing provided by Eklund (1994) [19], this is described in full by Hays (1995) [26] who offers newer more accurate adhesional force model description. The rapid growth of computer simulation research within this particular area has led to a complex, versatile and efficient models that can be adapted to other applications, forming the beginning of research toward other studies outside xerography. In

4

particular Oswah looked at this approach within filling processes with charged powder [47]. Many recent powder technology publications specific have been made by Eklund [19], Elsdon [20], Hays [29], Matsusaka et al. [41], Masuda et al. [42], Osawah [47] and Yoshida et al. [70] (particularly in regard to xerographic applications) which provide detailed inter-particle force models for experimental use. Overview of mechanics involved at microscopic scale and forces involved in particles is provided by Callister [11]. They also provide some overview of electrical effects. With regard to electric field application and electrostatic; Background electric theory discussion and concepts are provided by Purcell (1965) [48]. Applications to electrophotography (xerography) and specific physics involved is discussed by Schein (1992) [56] and Diamond (1991) [17]. Cho (1964) [13] outlines fundamental theory of induction charging of particles. Induction is also discussed by Wu (2003) [69] in terms of particle size, with various models of induction given. The effects involved within charge transport are outlined by Malave-Lopez and Peleg (1985) [39] in their mathematical linearization of electrostatic charging and decay curves in powders. The extremely complex modes of charged particle transport that exists in interacting collisional particle system within electric fields is discussed by Gartstein and Shaw (1999) [61], Melcher, Warren and Kotwall [43], Matsusaka and Masuda [42] and Schmidlin [57], (1989). Gartstein and Shaw [61] analyzed the transport mode of particles when contained within an electrostatic wave. They theorise and prove transport modes, giving models that can be used in simulation. The distributed particle velocities within an electrostatic wave were shown to vary between zero and wave phase velocity, and even become negative for some particles. Correlation of microscopic particle movement against average macroscopic values for particle transport and interaction is proven.

5

The different methods of contact charging are examined by Cho [13] and also more recently Matsusaka and Masuda (2003) [42] who provide three classifications of metal-metal, metal-insulator and insulator-insulator contact. Elsdon and Mitchell (1976) [20] describe rolling and sliding contact electrification effect in polymers. Yoshida [70] (2003) looks at similar aspects later on in research studies into the impact behaviour of polymers in contact charging to find a charging model particular to polymers. Other forces models also require their own review within this report. Most forces can be represented in particle forces during electrostatic sieving discussed by Bailey (1984) [9] in his discussion of electrostatic phenomena during powder handling. Hartley (1985) [25] discussed Van der Waals effect and its role in tensile strength with different packing density structures. VanderWaals effect is also discussed and compared against electrostatic forces by James Q. Feng, Dan A. Hays [22]. Feng Q. James [21] discussed the importance of electrostatic interaction between two particles and its importance in numerous industrial and natural processes and explained the electrostatic interaction between two charged dielectric spheres in contact. In his work two touching sphere of equal size and permittivity but arbitrary amount of charge is studied. He also explained the phenomenon of relative importance of electrostatic forces on powder particles. In electrophotography, electrostatics forces are utilized to move charge particles from one surface to another for the purpose of producing high quality print. Despite numerous studies conducted over decades results obtained from these were not consistent; the importance of electrostatics and non-electrostatics force is not completely explored and still a matter of debate among scientists. For example the results obtained by Donald and Watson [18], Goel and Spencer [23], Hays [27], Lee and Ayala [38], and Iimura et al. [35] indicate that toner adhesion is dominated by electrostatic forces, while the analyses of Krupp [37], Nebenzahl et al. [46], and Rimai et al. [53] appear to show that toner adhesion is mainly due to van der Waals. Some investigations show that both van der Waals and electrostatic forces can play an important role in adhesion [22].

6

The problem of two electrified spheres has been studied because of its importance in numerous industrial applications by many authors. Earlier authors considered conducting spheres because of relative simpler treatments in boundary conditions. The complete solution in bi spherical coordinates for two charged conducting sphere in an arbitrary uniform electric field was also obtained. The problem involving two charged spheres is not addressed in any literature until recently by Nakajima and Sato [45]. However these authors mainly Nakajima and Sato included two conducting sphere, a charged sphere near a grounded conducting plane, a charged sphere on a thick plane wall [45]. In chapter 3, we present detailed discussions on the inter-particle force models that form the basis for the current work.

7

CHAPTER 3: THEORITICAL BACKGROUNDIn this chapter, different inter-particle force equation in the presence of electric field will be discussed and an appropriate force model will be selected to calculate forceseparation distance relationship for toner particulate systems.

3.1 Force LawsThere are three different types of inter-particle forces that come into existence when a particle is subjected to external or internal force: contact force, short-range and long range. The contact force normally exists as repulsive force only it is due to the structural deformation of one particle intruding upon another particle. These forces are mechanical in nature and are always large compared to other forces.

-

Contact (overlap) Short-range Long-range

Figure 3.1 Different force laws The short -range forces are usually attractive even for the particle with like sign. The short-range force gets contribution from Van der Waals forces, electrostatic force and hydroscopic effects.

8

The long range forces are principally electrostatics force even drag force can be considered in this range. In this it is important to consider the effect of all charged particles not just the ones that are close but even present in some distance. Two particles of same mass and charge can have different adhesion due to its orientation [34]. To enable detailed analysis, the problem needs to be simplified by reducing the number of interacting particles. A relevant model should involve at least two spheres, because particles in powder or granular materials will make contact or may be at a distance from each other. Powder particle of narrow size can be manufactured. Considering two equal size spherical particles by ignoring effect of particle size distribution allows attention to be focused on effect of charge distribution among dielectric particles and the effect of permittivity of the dielectric particle material [21]. There are many views available when it comes in dealing with inter-particle forces in both dry and wet system. In physical terms, the most prominent forces in particle interactions are the adhesive (between a particle and a surface of a material) and cohesive (between particle of the same material) forces. When it comes to attractive forces present in air we see mainly the van der Waals forces and electrostatic forces [50].

3.2 van der Waals Forcevan der Waals force dominate in the contribution of adhesive force in polymeric materials, inorganic commercial powder and with similar adhesion properties in organic materials. The forceseparation is described by the van der Waals forces [50]. The occurrence of particle adhesion to substrate has great importance in technological fields of semiconductor, electrophotography, pharmaceuticals, paints, agriculture, aeronautics and space etc. The force needed to remove the particle from substrate depends on mechanical properties and the interaction potential existing between two materials [52]. Knowledge of adhesion between the solid or powder surface is important in understanding the sticking phenomena occurring in Microsystems. The van der Waals force originally arises by spontaneous polarization in electrically neutral materials. Polarization of atoms and molecules is inherent to all molecules and has very little dependence on external conditions. An instantaneous dipole that

9

occurs in one material can induce dipole in a nearby neighbouring material thus giving rise to a relatively short range van der Waals interactions. The magnitude of van der Waals forces can be highly sensitive to the microscopic substrate structures because of their short interaction range. Thus the effect of van der Waals forces .i.e. non-electrostatics short-range force can be reduced but cannot be totally ignored. The occurrence of van der Waals forces is related to the fundamental mechanism of the electrostatic forces but this force is normally regarded as non-electrostatics forces in adhesion and cohesion of particles. This type of interaction gives rise to surface forces [22]. In discussing particle adhesion/cohesion, the contribution of van der Waals forces is evaluated based on theoretical calculations. For the case in which a particle is bound to a substrate through surface forces, the force needed to remove that particle from the substrate is given by (Johnson, Kendall and Roberts) JKR theory as follows [53] Fs = 3 W AR * 2 (1)

Where W A represents the work of adhesion which is twice the surface energy between the interface . So therefore W A = 2 and negative force indicates that the applied force is in opposite direction of the applied force and R * is given as follows.

R* =

R1R2 R1 + R2

where R1 and R2 are radius of the two particle. For the case of particles of same size

R * will be reduced to half of R the radius of the particle .i.e. for the case of same sizeparticles the cohesive force would be half of the adhesive force between the particle and the planar substrate[53]. In this theory both tensile and compressive interactions contribute to the total contact radius of the particle. This model is derived using contact mechanics. This theory assumes that there is no long-range interaction [55]. Again the force needed to remove the particle from the substrate is given by and Derjaguin, Muller and Toporov) DMT theory which is represented as follows [50] Fs = 2W AR *

(2)

10

And all other parameters represent the same values as JKR theory [50]. The DMT theory originally generalizes the Derjaguin [52] concept of adhesion-induced deformation between the particle and the substrate and includes the tensile interaction to obtain the model. There has been a debate over many years up to the present about which model should be used. So we have considered both the theory for calculating the forceseparation model.

3.3 Electrostatic ForcesElectrostatics forces on powder and granular materials arise due to net electric charge on the particle or externally applied electric field. The charge can be acquired by several ways such as by contacting by other materials, by corona ions, induction charging, triboelectrification charge, etc. In general, electrostatics force becomes important when particle material is electrically insulting so the electric charge can be retained [22]. Electrostatic force is a long range force. It includes both repulsive and attractive force, which includes Coulomb force, fixed dipoles, patch charges and image charge. Many differentforce distance relationships are possible for different electrical effects and particle geometries. The Coulomb charging effect is considered the most important in particle behaviour. Fixed dipolar layer is of significance for reactive or moisture-coated surfaces. The patch charge produces much smaller electrical field which arises from work function in homogeneity over grains of overall neutrality [50].

3.4 ElectrophotographyTo aid the understanding of electrostatic and non-electrostatic force contribution to adhesion and cohesion, the phenomena of electrophotography is explained. The control placement of small, charged particles from one surface to another is a key requirement of electriphotographic making also known as xerography. Electrostatic and electrodynamic force or non-electrostatic force contributes towards the placement of this toner particles. The toner used in electrophotographic copiers is a cohesive powder with insulating particles consisting of pigment dispersed in a polymer resin. Toner particles are like

11

snow flakes they are in all shapes and size less than 10 m . Below (Figure 3.2) we can see the particle morphology of a toner particle [22].

Figure 3. 2 Particle morphology of toner particles The process begins by photoreceptor getting uniformly charged, typically using a grid controlled corona or a roller charger [54]. The photoreceptor is then image wise exposed using a writer that comprises either a laser scanner or LED array, thus an electrostatics latent image is created. A visible image is then created on the photoreceptor by bringing the electrostatic latent image into close proximity with electrostatically charged toner particles, and then these toner particles are attracted to and become deposited on the latent image bearing photoreceptor. The image is then transferred to a receiver material. This is the most important part of the electrophotography, subjecting the toner particles to an applied electric field and influencing them towards the receiver. The toned image is then fused and the photoreceptor is cleaned and made ready for the next job [22].

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Figure 3.3 Xerographic development The donor is a source of charge particles called toner which are moving under electric field to develop charged image on a receiver. The charge area is created by exposing a photoreceptor with laser as shown above (Figure3.3) [34]. In the absence of particle charge and externally applied electric field, electrostatic force disappears in powder particles. For particles without significant amount of electrostatic force van der Waals force becomes dominant. For electrophotographic application, the electrostatic force is utilized in different processes. A charged particle on a material surface is subjected to image force due to induced image charge in substrate. This image force attracts the particle towards the surface and so it is adhesive in nature. When the electric field is applied to detach the charged particle from the substrate, detaching and adhesive component of the electrostatic force must be considered. The applied electrical field also introduces an adhesive component of electrostatics force through the field-induced multipoles in the particle and their images in the residing surface. Knowledge of adhesion between the solid surfaces is important in placing or moving the toner particles to the required location which is contributed by van der Waals force.

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3.5 Typical Application involving electrostatic charging of particles.The phenomena of force-separation distance have its application in various fields of science and technology. Its application can be found in solid state batteries which uses the theory of piezoelectric generator action which will be explained in detailed, semiconductor industry, the adhesion of medicinal particles to the pharmaceutical industry, the controlled placement of small, charged particles from one surface to another in xerography, characterizing surface electrical properties, detecting defects of an integrated circuit and in biotechnological field [64].

3.5.1 Application of piezoelectric effect in solid state battery The voltage created by dipole movement changes when a mechanical compression or tension is applied on a poled ceramic material. It could be shown from the (Figure 3.5b) the compression along the direction of polarization or tension perpendicular to the direction of the polarization generates voltage of same polarity as the poling voltage. The voltage has a polarity that is in a direction opposite to that of the poling voltage as shown in (Figure 3.5c). The actions in which the piezoelectric element converts the mechanical energy of compression or tension into electrical energy or generator action is used in solid state batteries. The compressive stress and voltage generated is proportional to material specific stress. It alternatively stores electrical energy generated by piezoelectric element which has its application in solid state batteries [64].

a: disk after polarization (Poling)

b: disk compressed: generated voltage has same polarity as poling voltage

c: disk stretched: generated voltage has polarity opposite to that of poling voltage

Figure 3.4 Generator actions of a piezoelectric element

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3.6 Inter-particle force equationsIn electrophotography, electrostatic forces are utilized to move charged toner particles from one surface to another for the purpose of printing. The net electrostatic force on a charged particle in an applied electric field can be written in a form consisting of three components, FE = - Q2/ (16 0 R2) + QE - 0 R2 E2 [22] Where, Q E - net charge on the particle of radius R, - strength of externally applied electric field, - permittivity of the medium surrounding the particle, (3)

0

- dimensionless coefficients are function of many variables such as thedielectric constant of the particle, geometric configuration of electrodes. A charged particle on a material surface is subjected to an electrostatic image force due to the induced image charges in the surface. Qualitatively this electrostatic image force attracts the particle towards the surface and therefore is adhesive in nature. The first term in this equation represents the electrostatic adhesion due to the attraction between the net charge on the particle and its image charge in the substrate, the second represents for Coulomb force due to the external field acting on the particle charge, and the third is the electrostatic adhesion arising from the attraction between the field-induced dipole (polarization) in an uncharged dielectric particle in an electric field and its dipole image in the substrate [22]. An interparticle force model is presented for ac-dc current flow in a semi-insulating powder. The model applies to contacting particles as in packed and fluidized bed. The electrostatic model is based on simple lumped parameter circuit theory and correctly predicts the field-frequency trend observed for decreased bubble formation and increased bed height in a fluidized bed. General ac-dc force equation By combining both ac-dc force equations one can obtain the following equation [14].

1 + ( p )2 2 Fmax = C 0 d2 Ebk E 0 2 1 + ( b )

(4)

C - conducting particles at point contact

0 permittivity of free space15

E0 field strength d particle diameter

- is added to allow for the possibility that there is a field frequency dependence onthe force

b - bulk relaxation time in powder p - single particle relaxation timeRelaxation time ( ) [9] - When the particle is charged, charge transfer at particle contact redistributes itself over the particle surface by electrostatic force. The rate of the charge transfer is a function of the relaxation time of the material.

= - relaxation time - particle resistivity

- particle permittivityAn expression for the bulk charge relaxation time in the bed in terms of bulk bed properties is given as bulk relaxation time is given as b = p

- apparent bulk powder resistivity is needed for the evaluation of the chargerelaxation time and average bed current . The powder resistivity is measured in a packed bed as follows

( m) = (9.406 x 1048) E-0.69 T-17.71 (d)5.12 e-0.188RH (packed bed)

(5)

where E, T, d, RH are respectively the electric strength (V/m), temperature (k), particle diameter ( m), and relative humidity (%).In this equation the surface conduction of a particle is evident by the strong dependence of the bulk reisistivity on the relative humidity. The apparent bulk permittivity p of a bed voidage follows by analogy with the equation of bulk resistivity

p 3 p 0 =

0

p 2 1 0 p 3 + 0

(6)

p = kp (The relative dielectric constant of the particle). 0

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The single particle relaxation time is given as p = kp - relative dielectric constant of the particle,

(k

p

+ 2) 0 d 4 s

(7)

s - surface conductivity of the particle and is given as follows s =d d ln p rc (8)

p - the apparent resistivity of the particlerc - particle contact radius The following two equation will be used to relate the bulk bed resistivity and apparent (single) resistivity, is related to the corresponding single and contact particle resistances Rp and Rc as follows

d

= Rp+ Rc , Rp =

pd

1< < 2 for a conducting spherical particle contacting a wall or another particle and Ebk is a limiting field strength such as is given by the Fowler-Nordheim equation for high field emission. The limit ~ 1 infers a field limited breakdown without deformation at the point contact. The case = 2 and C = 1.37 applies to a point contact. Intermediate values of between 1 and 2 occur for a Hertzian deformation at the contact area. For glass particles gives an experimental value of less than unity ( = 0.656) evaluated at the interface of an electric suspension. For conducting spherical particles in contact with a conducting flat surface Colver [14] reports experimentally = 2 and C = 1.24 1.33 with an average value of 1.29. For a resistive sphere and a conductive plane, Arteaga et al. [1] suggests a model with = 1 1.2 for 1.9- and 3.7-mm spheres. A constant has been added in this to allow for the possibility that there is a fieldfrequency dependence on the force. For example = 1 implies that both the breakdown field Ebk and the field E0 in the equation conform to the simple RC frequency model where as for = /2 there is a field frequency dependence with Ebk taken as being constant over the cycle. These possibilities are purely speculative as an experimental value for has yet to be determined [14].

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During powder handling operations, particles make frequent contact with surfaces often metallic, and become electrically charged due to the process of contact electrification also known as triboelectrification. When a particle charges during handling the charge transferred at the particle contact point will redistribute itself over the particle surface by electrostatic forces. The rate at which the redistribution proceeds depends upon the electrical relaxation time of the particle material. Relaxation time is given by the product where = particle resistivity and = particle permittivity [69]. The expression for the force on an ion in presence of current has also been derived from first principles without the assumption about its conservative character. Here we start from the expression for force in the presence of current. First force expression comes about as the classical limit of Ehrenfests theorem applied to the rate of change of ionic momentum. Formally this limit corresponds to high ionic masses or high kinetic energies. Another approach for force equation starts from an appropriate quantum-classical Lagrangian [66]. The force expression then results from the Euler - Lagrange equation of motion for the classical ions. Both approach yield the same expression for the force on an ion, with position R, due to the self consistent electronic density (r) under current flow is the following expression F=-

dr r (r)

v

(9)

Which can be equivalently written as F=-

i

i

v i R

- lim

0

dE

v R

(10)

Where v - is the electron - ion interaction. The first term on the right hand side of the equation (10) is similar to the usual Hellmann Feynmann contribution to the force due to localized electronic states i . The second term is the contribution due to continuum states. The wave function is eigen differentials for each energy in the continuum [66]. A mathematical expression for the electric force acting on a sphere placed in a nonuniform electric field is derived taking into account conductivity and permittivity of both sphere and medium as well as the initial velocity and charge of the spherical particle. The expression demonstrates a time dependent variation of the force in a non-uniform field.

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For a uniformly polarized sphere the external field coincides with the field of dipole with dipole moment p = VP where V is the volume of the sphere P is the polarization vector. The electric force acting on a polarized sphere is given by the following equation

F j = Pkk =1

3

E0 k x j

(11)

where F - force (N) The component of the polarization vector Pk can be expressed in terms of field

E*k inside the sphere by the following equationE*k = E0k where the field strength inside and outside the sphere is designated as E* and Ee . Eok - projection of vector Eo on Cartesian co-ordinate x1, x2, x3 (Vm-1), calculated for spheres centre, Eo - undisturbed electric field strength (Vm-1) E*k - projection of vector E* on Cartesian coordinate x1, x2, x3 (Vm-1), E*, Ee - electric field strengths inside and outside sphere (Vm-1),

Pk

3 0 e

(12)

P, Pk - polarization vector and its projection Cartesian coordinates (Cm-2) , p - dipole moment (Cm-1),

0 - electric constant (Fm-1), i , e - permittivity inside and outside the sphere.The final formula for the components of the total force acting on a sphere has the form Fj = Q E0f + 3 V 0 e where Q is the free charge carried by the particle and V is the volume of the sphere. In view of the fact that the last term of the equation (13) determining E*, is proportional to both field strength gradient

k =1

3

( Eok-- E*k )

E0 k x j

(13)

E0 k and to particle velocity xj the force x j

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(13) depends both on the electrical parameters of the particles and medium and on the particle velocity [44]. The electrophotographic technology requires the careful control of adhesive, cohesive, and electrostatic forces in order to properly and repeatedly position micrometer- sized pixels, with the output being a hard copy of a digital image. Electrostatic force arises from the fact that the toner particles are charged. Electrophotography produces documents and images by precisely placing large number of micrometer size pigmented particles generally called toner particles. The force produced is long range and are significant in allowing the toner particles to be deposited on the charged photoreceptor. They also contribute to the adhesion of the particle to the photoconductor, as well as columbic repulsion between toner particles. Electrostatic forces are generally used to transfer the toned image from the photoreceptor to the final receiver. Electrostatic forces between a uniformly charged spherical particle of radius R and charge q and its image charge in a grounded conducting plane is given by the following equation

1 q F= 4 0 2 R where

2

(14)

0 - is the permittivity of the free space.The toner particle could not be electrostatically detached from photoreceptor by applying electrostatic field alone. So accordingly the force FI adhering a toner particle of radius R to the photoreceptor is calculated using the method of images as follows

2 R 2 FI = 0

(15)

Where 0 is the permittivity of free space, the particle charge is approximated as varying with the surface area and having surface charge density constant. The applied detachment force FE is given by FE = 4 R 2E Where E is limited by the paschen discharge strength of air. The type of force that adheres toner particles to the photoreceptor arises from electrodynamic interactions. An instantaneous dipole that occurs in one material can (16)

20

induce its dipole in neighbouring material, this gives rise to short-range Van der Waals interactions. The force Fs which is required to remove that particle from the substrate for a particle which is bounded to a substrate is given by JKR theory (1) and DMT theory (2) [54].The JKR theory states that both tensile and compressive interactions contribute to the total contact radius. This model is derived using contact mechanics. Here it is assume that there is no long-range force and DMT theory takes different view of how and where the adhesive force acts [55]. The effect of many body interactions on the electrostatic force between colloidal particles in a charged pore is being solved by using non-linear Poisson equation. The electrostatic force acting on a sphere is calculated by integrating the Maxwell stress tensor over a suitable surface. The electrostatic interaction force F between two identical sphere acting along the line of centre is obtained by integrating the stress tensor over the mid-plane as in equation.

kT * F = F e spheres over a mid-plane has the following form:

2

(17)

The dimensionless force F* in a monovalent electrolyte system between two identical

F*=

[2(cosh 1) + 0

2

]RdR

(18)

where, R is the radial coordinate along the midplane . The dimensionless co-ordinate is defined as follows R=kr, Z= kz. The radial and axial co-ordinate is r and z, e is the electronic charge, is the space potential, k is the Boltzmann Constant, T is the absolute temperature and k is the Debye parameter which has the following form,

e2 k= zi2 n i kT i where,

1

2

2e 2 N AC (z + + z ) = kT

1

2

(19)

is the permittivity of the electrolyte, NA is Avogadros number C is theconcentration of the electrolyte, scaled by the Debye length K-1, and the integrand is evaluated over the mid-plane. For the case of two spheres situated axisymmetrically in a pore, the integral in equation (18) is performed over the portion of the mid-plane within the pore, which limits the radial integration to the distance of the pore wall from the axis of symmetry [58]. The electrostatic interaction between two charged dielectric spherical particles in contact is taken here for the description of the

21

behaviour of powder or granular material that consists of electrically insulating particles [21]. In this we consider two touching dielectric spheres, namely sphere 1 and sphere 2 of same radius and same permittivity , carrying charges Q1 and Q2 in a dielectric surrounding medium of permittivity 0 . The electrostatic force can be expressed as sum of three lumped terms as follows F(Q) = Q2 - Q + (20)

The first term is due to the attraction from the image charge Q induced in sphere 2. The second term represents the Coulomb force from the interaction between Q and the electric field generated by the sphere 2. The third term describes the fact that a net dielectrophoretic force can be induced by the electric field from the charge on sphere 2 even when sphere 1 does not carry net charge (i.e., Q = 0). In general, the coefficients , , can be functions of the particle permittivity and geometric configuration such as the particle size ratio as well as the distance between particles[21]. In electrophotography, charged toner particles are moved from one surface to another by applied electric field. The phenomenon of triboelectricity is widely used to charge toner. Efficient electric field transfer of toner particles between surfaces is important in several process steps of electrophotography. The applied force on a charged particle due to an external electric field E is given by the equation Fa = (K , f m ) QE (K , f m ) 0 R2E2 where, (21)

(K , f m ) and (K , f m ) are polarization correction factors. ParticleQ2

detachment occur when Fa is equal to adhesive force Fad . Fad = FNE + (K , f m )

16 0 R 2

(22)

where, Q is the particle charge, R is the particle radius, 0 is the permittivity of the free space and (K , f m ) is the particle polarization correction factor that depends on the particle dielectric constant k and the fraction of monolayer coverage f m . If a nonelectrostatic adhesion component such as VanderWaals force FNE is included, the particle adhesion is given by the equation (22). The electrostatic adhesion

22

between a uniformly charged dielectric sphere and conductive substrate can be described by an electrostatic image force given by this equation (23) Fi = ( K , f m )

Q2

16 0 R 2

[26].

(23)

The electrostatic force of adhesion acting on a charged insulating sphere in contact with a conductive plane has important applications in many different fields such as electrophotography, semiconductors, atomic force microscopy, and micro-tribology. For example, in electrophotography, a triboelectrically charged toner particle must be efficiently developed on a photoreceptor and transferred to paper using electric field. The charge q in each charge point is the total charge on the sphere Q divided by the number of charge points K, q=

Q Q = K 4N 2

(24)

where, the total number of charge points K can be derived by summing of the charge points all of the annuli as follows K=N 1 i =0

k

i

= 2N

N

0

sin x

N ( N )dx 4

2

(25)

The number of charge points on the first annulus nearest the conductive plane is given, in the limit for a large N, by k0 = 2 N sin ( / 2 N )

4N 2

(26)

The plane of the i th annulus across the z axis at z i is given as z i = R 1 cos

i + , where i = 0. 2 N N

(27)

Using the first two terms of the Taylors series expansion, the separation z0 of the first annulus from the reference plane z = 0 is given as follows,

R z0 = 2 2N

2

(28)

Consider the electrostatic forces due to the interactions between the charge points located in proximity to the conductive plane with their image charge located symmetrically across the conductive plane. Using Coulombs law, the force on a single charge point q in the first annuli due to its own image charge point F i located symmetrically across the conductive plane can be given by the following equation

23

Fi =

1 Q2 4 q2 = 4 0 (2 R )2 2 4 0 (2 z 0 )21

(29)

The functional dependence of q and Z0 on the number of annuli N as given by equation (24) and equation (28) cancels out. Since there are approximately charge points in the first annuli equation (29), the contribution to the electrostatic force by these charges, which is called as proximity force Fp given as equation (30)

Fp = F =

Q2 4 4 0 (2 R )2

1

(30)

where, Q is the total charge on the sphere of radius R and permittivity 0 . Since the number of point charges considered in the proximity annulus is much smaller than the total number of charge points, i.e. k0 1cm) at Q = 2 0 R 2 E ( 1 + 2

) with relative permittivity 0 = 8.854 10 12 Nm2/C2, the

separation distance between the particle r is varied with distance less than and equal to the particle radius .i.e. r > R. and the work of adhesion W A is given as 0.08 J/m2 between the two toner particle [22]. Figure 4.5 shows that the force-separation relation for toner particles computed based on equation (33).

Force vs separation distance (Type-1)2.0E-05 1.5E-05 Force (N) 1.0E-05 5.0E-06 0.0E+00 0.0E+00 -5.0E-06 1.0E-06 2.0E-06 3.0E-06 4.0E-06 5.0E-06 6.0E-06

separation distance (m)

Figure 4.5 Electrostatic force versus separation ( for type-1 simulations explained later) However, in this study, we considered a range of inter-particle force-separation scenarios to assess the different force contributions as explained later in Section 4.6.

4.6 Simulation ExperimentThe discrete particle simulation (DPS) which is identical to the discrete element method (DEM) without the electrostatic contribution, which was originally developed by Cundall and Strack [15] is used in the present study. The advantage of using DPS to study the materials is its ability to give more information about micromechanical characteristics of large particle system during mechanical loading. The DPS code models the interaction between the contiguous particles as dynamic process and the time evolution of the particles is advanced using an explicit finite difference scheme.

35

A simple force mechanism was employed between the particles in contact: normal and tangential contact springs were assigned, and slipping between particles would occur whenever the (specified) contact friction was attained [8]. All the simulations were carried out on a Sunblade 2000 workstation (typically one case of a simulation requires about 3 week to run the programme). All the assemblies contained about 2000 mono-dispersed particles in dense packing and the individual shape of the particle is spherical. The initial assemblies were isotropic and homogeneous. The properties of the particles assigned to the granular samples are shown below in table 4.1. Material Number of particles Initial void ratio Initial solid fraction Initial porosity Initial number of contacts Coefficient of inter-particle friction () Friction w Normal spring stiffness (Kn), N/m (109) Tangential spring constant (Kt), N/m (109) Density of particle ( p ), kg/m3 (103) Size m (10-6) Work of adhesion (WA), J/m2 Toner 2,000 0.5358 0.6511 0.3488 5151 0.5, 0.25, 0.01 0.5 1 1 1 10 0.08

Table 4.1: Properties of the toner particle

The assemblies were subjected to tri-axial mechanical loading condition along their boundaries (33 > 22 = 11) (Figure 4.6).

During the Tri-axial compression simulations, the height of a three dimensional assembly was slowly reduced at a constant rate along the 3-3 direction, while maintaining a constant stress in the 1-1 and 2-2 directions as shown below in Figure 4.6.

36

3

1 2

Figure 4.6: Stress tensor diagram (3D) The above diagram shows a cube subjected to tri-axial external forces. The forces on each face can be resolved into components, which are parallel to the cubes three axes. The nine components can be arranged into the stress tensor. The axial was advanced in small increments of 33 = 5 x 10-3, and several relaxations steps were performed within each increment. In analysing the results of the simulations, the bulk characteristics of a material assembly are computed from the micro-scale data of its particle and contacts. The average stress tensor ij in an assembly can be directly computed as a sum of dyadic products associated with its M contacts [15]:

ij =

1 V

lxy

xy

f

xy

(38)

where V is the assembly volume. Each product is for a contact xy between particles x and y, and the pair xy is an element in the set M of all contacts. The branch vector

l xy connects a reference point on a particle x to a reference point on particle y; and

f

xy

is the contact force exerted by y on x . Each vector can be expressed as the

product of scalar magnitudes and unit directions as [6, 15]:

37

ij =

1 V

lxy

xy

[f

xy

( mixy n xy ) + f j

xy ,t

( mixy t xy ) ] j

(39)

where nxy is the outward unit normal of particle x at contact xy, txy is the unit tangential vector aligned with the tangential component of contact force fxy,t, and mxy and l xy are the direction and length of branch vector lxy: f xy = fxy , n

nxy + f

xy ,t

txy

(40)

lxy = l xy mxy The average stress could be approximated by neglecting the tangential forces as [52]:

ij =

1 V

xyM

l

xy

f xy ijxy

(41)

xy xy xy with a contact fabric ij = mi n j associated with each xy contact, Equation (41)

suggests that stress and fabric may be closely correlated, but our results demonstrate later that this correlation exists in a compacted bed only if the weekxy contacts are excluded from the calculation of fabric < ij >, a finding in agreement

with other recent works for granular assemblies subjected to quasi-static shearing [6,xy 50, 65]. The fabric tensor ij of N contacts can be written as:

= (1 / N ) nixy n xy jxy ij 1

N

(42)

Sxy The fabric tensor ij can be written as:

ijSxy = (1 / N ) nixy n xy jkS

(43a)

where the sent of contacts S is the set of contacts k that have a force magnitude |fk| greater than average f : S = { k: |fk| > f } (43b)

Ns is the number of contacts in S.

4.6.1 Simulation Types:For comparison purpose the simulation are run for six types of force model for 3 different types of inter-particle coefficient of friction () = 0.5, 0.25, and 0.01. The types of force are shown in tabular column.

38

Simulation Types

Inter Particle Forces Considered Electrostatic Force Variation in short range force (Term 1) (Term 2) (Term 3) Long range force Long range force JKR (Term 4 a) DMT (TERM 4 b) Adhesion (pull-off) force (Non-Electrostatic Force) Mechanical contact force (without short range and long range contributions)

Type 1 Type 2 Type 3 Type 4 Type 5 Type 6

= Accounted, = Ignored in calculating the net inter-particle force vector Table 4.2. Contribution of force term in the simulations (*Terms are explained in the next page)

39

These are the different types of inter-particle force models which are used for simulations. Term1: This term accounts the variation in the short-range electrostatic force (as a function of separation distance between the particles). Existing studies in this field often ignore this variation in short-range force contribution.

FI =

Q2

16 0 r 2

(44)

Term 2: This term accounts the long-range contribution of electrostatic force due to the external field acting on the particle charge.

FE = Q E .

(45)

Term 3: This term accounts for long-range force contribution of electrostatic force due to electrostatic adhesion arising from the attraction between the field induced dipole in an uncharged particle in an electric field and its dipole image in the substrate.

FE = 0 R 2 E 2Term 4 a: This term represents the Adhesion (pull-off) force due to JKR theory. Fs = 3 W AR * 2 Term 4 b: This term represents the Adhesion (pull-off) force due to DMT theory Fs = 2W AR *

(46)

(47)

(48)

40

Chapter 5: Results and discussionsFigure 5.1 shows the evolution of the axial stress ratio 33 / 11 during the tri-axial compression test. We observe that the variations in the short-range electrostatic forces significantly influence the value of axial stress ratio. If we account the contribution of the variation in short-range electrostatic force type (1, 4) then there is an increase in the peak value of the macroscopic stress ratio 33/ 11by about 18% for the inter-particle coefficient of friction () 0.5, compared with the simulation that ignores this contribution (types 2, 3, 5, 6). Similarly the increase in stress ratio for systems with = 0.25 the axial stress ratio is about 30% and for = 0.01 it is approximately 73% higher. We observe that identical test results are obtained for the stress ratio in accounting pull-off force, calculated using JKR and DMT theories (types 1, 4).The test results show that, if we ignore the variation in short-range electrostatics forces between the particles, then values of macroscopic stress ratio

33 / 11 between the cases that account the long-range electrostatic contribution(types 2, 5) and that ignore it types (3,6) remain fairly the same. Thus the results clearly show that accounting the variation in the short-range electrostatics force has the most significant influence on the bulk stress ratio 33 / 11 developed in the assemblies. For clarity purpose, in Figure 5.2 we show that the evolution of 33 / 11 for assemblies with different values of inter-particle friction. It is evident from Figure 5.2 that a decrease in the value of inter particle friction coefficient ( ) results in a decrease in the value of macroscopic axial stress ratio in all types of simulations considered in this study. Generally the value of 33 / 11 is non-uniform (post-peak softening) for the entire stages of compressive strain regime. However the extent of this non-uniformity drops down when the value of inter-particle friction coefficient reduces. For assemblies having the lowest value of inter-particle friction considered here ( = 0.01) the presence of short-range electrostatics force (type 1) leads to pronounced peak value in the axial stress ratio followed by a post peak soften of the material. This trend is absent if the variation in short range electrostatics forces are ignored in the systems (types 2, 3).

41

Axial Strain vs 33/11 ( = 0. 5)5.00 4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0.00

33/11

0.05 type1

0.10 type2

0.15 Axial Strain type3 type4

0.20 type5

0.25 type6

0.30

Axial Strain vs 33/11 ( =0. 25)5.00 4.50 4.00 3.50 33/11 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0.00 0.05 0.10 0.15 Axial Strain type1 type2 type3 type4 type5 type6 0.20 0.25 0.30

Axial Strain vs 33/11 ( = 0.01)5.00 4.50 4.00 3.50 33/11 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0.00 0.05 0.10 0.15 Axial Strain Type1 Type2 Type3 Type4 Type5 Type6 0.20 0.25 0.30

Figure 5.1 Variation of the axial stress ratio during compression

42

Axial Strain v s 33/11

5.00 33/11 4.00 3.00 2.00 1.00 0.00 0.00 0.05 0.10 0.15 Axial Strain type1( = 0.5) type3( = 0.25) type2( = 0.5) type1( = 0.01) type3( = 0.5) type2( = 0.01) type1( = 0.25) type3( = 0.01 type2( = 0.25) 0.20 0.25 0.30

Figure 5.2 Variation of axial stress ratio during compression, showing the effect of inter-particle friction

From Figures 5.1 and 5.2 we can infer that, among the simulations for the test material and test conditions considered here, JKR and DMT theory do not have any significant differences in the results, assemblies which for all values of inter-particle friction.

In Figure 5.3, the macroscopic friction is presented in terms of Sin =

1 3 1 + 3

during compression. From figure 5.3 we observe that the presence of short-range electrostatics force significantly influence the bulk friction of the assembly. By accounting the contribution of short-range electrostatics force (type 1, 4) there is an increase in the peak-value of the macroscopic Sin by 20%-50%, when compared with the results for the assemblies that do not consider this contribution. The test results we obtained through simulation infer that if we ignore the variations in the short-range electrostatics force between the particles, then the value of macroscopic Sin between cases that account long range electrostatics contribution (types 2, 5) and that ignore the long range electrostatics force (types 3, 6) remains fairly the same. From Figure 5.4 it is evident that by decreasing the value of inter-particle friction, the assemblies present a decrease in the value of the macroscopic friction value in all types of simulations considered in this study.

43

Axial Strain vs Sin ( = 0.5)0.70 0.60 0.50 Sin 0.40 0.30 0.20 0.10 0.00 0.00

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Axial Strain vs Sin ( = 0.25)0.70 0.60 0.50 Sin 0.40 0.30 0.20 0.10 0.00 0.00

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Axial Strain vs Sin ( = 0.01)0.70 0.60 0.50 Sin 0.40 0.30 0.20 0.10 0.00 0.00

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Figure 5.3 Variation of the macroscopic friction (Sin ) ratio during compression.

44

Axial Strain vs Sin

0.70 0.60 0.50 Sin 0.40 0.30 0.20 0.10 0.00 0.00 0.05 0.10 0.15 Axial Strain type1( = 0.5) type3( = 0.25) type2( = 0.5) type1( = 0.01) type3( = 0.5) type2( = 0.01) type1( = 0.25) type3( = 0.01) type2( = 0.25) 0.20 0.25 0.30

Figure 5.4 Variation of macroscopic friction (Sin ) ratio during compression with the effect of inter-particle friction

Generally the value of the Sin is non-uniform for the system. However we can see that the non-uniformity decreases as the value of inter-particle coefficient decreases. The system having the lowest inter-particle friction ( =0.01) considering the presence of short-range electrostatic forces .i.e. type 1 leads to a pronounced peak value in the Sin ratio followed by a post-peak soften of the material. This trend is absent if the variation in short-range forces are ignored. Figure 5.5 Shows a measure of the shear induced in the assembly during compression, presented in terms of the ratio of deviator to hydrostatic stress q/p (q =

33 11 , p = kk / 3) . In this figure we can observe that the presence of short-rangeelectrostatic forces plays an important role as it strongly influences the value of deviator stress ratio. All the general characteristics in the q/p curves are fairly identical to the macroscopic friction (Sin ) curves as one would normally expect.

45

Axial Strain vs q/p ( = 0.5)

1.80 1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00

q/p

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Axial Strain vs q/p ( = 0.25)1.80 1.60 1.40 1.20 q/p 1.00 0.80 0.60 0.40 0.20 0.00 0.00 0.05 0.10 0.15 Axial Strain type1 type2 type3 type4 type5 type6 0.20 0.25 0.30

Axial Strain vs q/p ( = 0.01)1.80 1.60 1.40 1.20q/p

1.00 0.80 0.60 0.40 0.20 0.00 0.00 0.05 0.10 0.15Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Figure 5.5 Variation of deviator stress ratio (q/p) during compression

46

Axial strain vs contacts ( = 0.5)6000.0 5000.0 4000.0 3000.0 2000.0 1000.0 0.0 0.00

contacts

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Axial Strain vs contacts ( = 0.25)6000.0 5000.0 4000.0 3000.0 2000.0 1000.0 0.0 0.00

contacts

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1mup25

type2

type3

type4

type5

type6

Axial Strain vs Contacts ( = 0.01)6000.0 5000.0 4000.0 3000.0 2000.0 1000.0 0.0 0.00

Contacts

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Figure 5.6 Variation of total number of contact during compression.

47

Figure 5.6 shows the evolution of geometric characteristics of the assemblies during compression, presented in terms of the total number of contacts. The results show that if we ignore the contribution of the variations in the short-range electrostatic forces (types 2, 3, 5, 6), then generally the systems presents a higher value of the total number of contacts. It should be noted that the nature of short-range electrostatic forces are adhesive and hence ignoring the variations in this force contributions leads to over estimating the number of contacts in the systems. However, this trend is less severe in the case of systems with high inter-particle friction value ( = 0.5) than the ones that have lower values of inter-particle friction. It is also evident that among systems that accounts the contribution of JKR and DMT pull-off force contribution, presents no significant difference in the results for the total number of contacts. Further, we can also notice that, systems that account only the long-range contribution of the electrostatic forces (types 2, 5) and the systems that account only the JKR/DMT pull-off forces (types 3, 6) present identical results with regard to the total number of contacts in the systems. It is evident that a decrease in the value of the inter-particle friction results an increase in the value of the total number of contacts in the systems, which is seen in all types of simulations considered in this study. Generally the value of total number of contacts considered here is non-uniform at the early stage of compression (pre-peak region), and thereafter remains fairly the uniform until the end of compression. Figure 5.7 represents the geometric characteristics of the ratio of sliding contacts (ratio of number of contacts sliding to the total number of contacts at a given strain). Generally, ignoring the contribution of the variations in the short-range electrostatic forces present, less fluctuations in the variation of the ratio of sliding contacts. Other features of the variations of sliding contacts are fairly similar to the results (Figure-6) on the variations in the total number of contacts. However, the difference in the value of sliding ratio between systems that consider the electrostatic forces (types1, 2, 4 & 5) and that ignores this (types 3, 6) generally tend to diminish if the inter-particle friction between the particles is low ( = 0.01).

48

Axial Strain vs ratio of sliding contacts ( = 0.5)1.20 ratio of sliding contacts 1.00 0.80 0.60 0.40 0.20 0.00 0.00

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Axial Strain vs ratio of sliding contacts ( = 0.25)1.20 ratio of sliding contacts 1.00 0.80 0.60 0.40 0.20 0.00 0.00

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Axia Strain vs ratio of sliding contacts ( = 0.01)1.20 ratio of sliding contacts 1.00 0.80 0.60 0.40 0.20 0.00 0.00

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Figure 5.7 Variation of sliding contacts ratio during compression

49

Axial Strain vs ratio of sliding contacts (JKR)1.20 ratio of sliding contacts 1.00 0.80 0.60 0.40 0.20 0.00 0.00

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1( = 0.5) type3( = 0.25)

type2( = 0.5) type1( = 0.01)

type3( = 0.5) type2( = 0.01)

type1( = 0.25) type3( = 0.01)

type2( =0. 25)

Figure 5.8 Variation of sliding contact ratio during compression with the effect of inter- particle friction. Figure 5.8 shows that for identical test conditions, a decrease in the values of interparticle friction results in a higher value of sliding ratio during compression. From the above results we observe that the role of short-range electrostatic forces significantly affects the macroscopic features (e.g. Macroscopic friction Sin , shear strength q/p and axial stress ratio 33/ 11 all of these present higher values if we account the short-range (adhesive) electrostatic forces when compared with systems that ignore the contribution) and the microscopic characteristics (total number of contacts and the sliding ratio decreases if we ignore the short-range electrostatic force contribution). Further, the above mentioned macroscopic parameters tend to be inversely proportional to the microscopic parameters. To understand the reasons for this characteristic, we analyse the nature of force networks develop in the granular systems considered here.

Figure 5.9 shows the variation of fabric anisotropy (33/11) by accounting all the contacts in the assemblies during compression (equation 42), and Figure 5.10 shows this measure by accounting the strong contact only (43a, b). The strong contacts are defined as the contacts carrying forces greater than the average normal contact in the assembly at a given strain level force.

50

Axial Strain vs 33/11 ( = 0.5)2.00 1.80 1.60 1.40 33/11 1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00 0.05 0.10 0.15 Axial Strain type1 type2 type3 type4 type5 type6 0.20 0.25 0.30

Axial Strain vs 33/11 ( = 0.25)2.00 1.80 1.60 1.40 33/11 1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00 0.05 0.10 0.15 Axial Strain type1 type2 type3 type4 type5 type6 0.20 0.25 0.30

Axial Strain vs 33/11 ( = 0.01)2.00 1.80 1.60 1.40 33/11 1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00 0.05 0.10 0.15 Axial Strain type1 type2 type3 type4 type5 type6 0.20 0.25 0.30

Figure 5.9 Variation of fabric anisotropy by accounting all the contacts in the assemblies during compression.

51

Axial Strain vs s33/s11 ( = 0.5)6.00 5.00 s33/s11 4.00 3.00 2.00 1.00 0.00 0.00

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Axial Strain vs s33/s11 ( = 0.25)6.00 5.00 s33/s11 4.00 3.00 2.00 1.00 0.00 0.00

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Axial Strain vs s33/s11( = 0.01)6.00 5.00 s33/s11 4.00 3.00 2.00 1.00 0.00 0.00

0.05

0.10

0.15 Axial Strain

0.20

0.25

0.30

type1

type2

type3

type4

type5

type6

Figure 5.10 Variation of strong contacts during compression

52

We observe that the general trend in the variation of these fabric anisotropy (of total and strong contacts) are identical with the variation of the macroscopic parameters (q/p, Sin , 33/ 11 curves presented in Figure 5.1, 5.3 and 5.5).However the comparisons between the Figure 5.9 and 5.10 show that the magnitude of the anisotropy of strong contacts are much higher than the anisotropy accounting all the contacts present. Existing investigations performed by Antony and Co-workers [2, 3, 4, 5, 7 and 64] on the force transmissions characteristics in sphere and non-sphere granular systems clearly indicate that the macroscopic strength characteristics of granular systems strongly depend on the systems ability to build-up a strongly anisotropic fabric structure of strong contacts.

Axial strain vs q/p, 1/2*sqrt( s33/s11)2.50 2.00 1/2* sqrt ( s33/s11) 2.00 1.50 1.50 q/p 1.00 0.50 0.00 0.00 1.00

0.50 0.00 0.30

0.05

0.10

0.15 Axial Strain

0.20

0.25

Axial strain vs q/p

Axial Strain vs s33/s11

Figure 5.11 Variation of fabric anisotropy of the strong contacts

Figure 5.11 shows the typical Correlation between the macroscopic strength curve q/p and the fabric anisotropy of the strong contacts alone. We also confirm that an excellent arrangement between the macroscopic strength and the constant anisotropy exists at the microscopic level satisfying:

q/p=1/2

s33/s11

(44

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Chapter 6: Conclusions and RecommendationAn attempt has been made to investigate the microscopic and macroscopic characteristics of the electrostatically charged granular systems subjected to mechanical tri-axial compression. We identify appropriate force-displacement relations to represent the combined mechanical and electrical loading conditions: the total forces acting between the particles are divided into several sub-groups, namely, short-range forces (with and without accounting the variations in them), long-range forces and the forces due to mechanical overlap between the particles. The toner particulate assemblies contain spherical particles in a dense packing. The study focussed on several properties of the assemblies at the macroscopic and particulate level. The results clearly show that accounting the contribution of the variation in the short-range electrostatic forces is the most significant influencing factor in determining the macroscopic and microscopic features of the assemblies considered here. The simulation results show that the role of short-range electrostatic forces significantly affects the macroscopic features (e.g. Macroscopic friction Sin , shear strength q/p and axial stress ratio 33/ 11 all of these present higher values if we account the short-range (adhesive) electrostatic forces when compared with systems that ignore the contribution) and the microscopic characteristics (total number of contacts and the sliding ratio decreases if we ignore the short-range electrostatic force contribution).The simulations shows strong link between the macroscopic stress tensor and the fabric tensor of strong contacts in all the types of models considered: The short-range electrostatic forces contributed to establish a strongly anisotropic fabric structure of strong contacts, thus leading to establish higher values of the macroscopic strength parameters. We also conclude that, for toner particulate systems considered here, no significant difference in the results are obtained to calculate the pull-off forces predicted by JKR and DMT theories, both the models yielded the identical results. The role of inter-particle friction is probed in detail in this study. A decrease in the value of inter-particle friction results in the systems to be viable to build strongly anisotropic strong force networks. This is further evident in the results of sliding ratio: systems with low inter-particle friction shows that sliding of contact is far more dominant in such systems. The systems that ignore the variation in short-range forces tend to present a more uniform variation in the ratio of sliding

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contacts and this trend is more dominant in systems having low value of inter-particle friction. Existing studies often ignore the variation in short-range electrostatic forces. Further, the analysis presented in this thesis is entirely new to the best of our knowledge and well deserves an effort to study. However future studies are required to fully understand the complexities present in charged particulate systems subjected to mechanical loading and a few are mentioned below: I. The particulates could be considered as poly-dispersed in size. II. The simulations need account more realistic (non-sphere) shape of the particles, which could prove to be very significant factor in this area of research. III. Electrostatic charges are assumed to transfer uniformly within the sphere shape particles. More realistic predictions should account non-uniform distribution of charge, even within a particle level. IV. The heterogeneity characteristics of force distribution in the charged particulate systems need to be quantified well in future. V. Future studies should account the scale-up effects in charged particulate systems. VI. The present study predominantly considered the bulk strong characteristics of charged particle systems. However, further analysis is required to present the distribution of the electric charge/potential within the particulate assemblies as required in Micro-Electro-Mechanical Systems (MEMS) systems.

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List of captions:Figure 3.1 Different force laws. Figure 3.2 Particle morphology of toner particles. Figure 3.3 Xerographic developments. Figure 3.4 Generator actions of a piezoelectric element. Figure 3.5 Two spherical particles of same size and charge separated by a distance, resting on electrically conducting substrate in a detaching electric field. Figure 4.1 Simulated image structure of particles. Figure 4.2 The principal steps in setting up a computer experiment. Figure 4.3 Role of a mathematical model. Figure 4.4 The timestep loop of the DPS algorithm. Figure 4.5 Electrostatics force versus separation ( for type-1 simulations explained later). Figure 4.6 Stress tensor diagram (3D). Figure 5.1 Variation of the axial stress ratio during compression. Figure 5.2 Variation of axial stress ratio during compression, showing the effect of inter-particle friction. Figure 5.3 Variation of the macroscopic friction (Sin ) ratio during Compression. Figure 5.4 Variation of macroscopic friction (Sin ) ratio during compression with the effect of inter-particle friction. Figure 5.5 Variation of stress ratio (q/p) during compression Figure 5.6 Variation of total number of contact during compression Figure 5.7 Variation of sliding contacts ratio during compression. Figure 5.8 Variation of sliding contact ratio during compression with the effect of inter- particle friction. Figure 5.9 Variation of fabric anisotropy by accounting all the contacts in the assemblies during compression. Figure 5.10 Variation of strong contacts during compression.

Figure 5.11 Variation of fabric anisotropy of the strong contacts.Table 4.1 Table 4.2 Properties of the toner particle. Contribution of force term in the simulations

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NomenclatureQ E Net charge on the particle Strength of externally applied electric field, Permittivity of the medium surrounding the particle, Field strength Particle diameter Bulk relaxation time in powder Single particle relaxation time Relaxation time Particle resistivity Particle permittivity Apparent bulk powder resistivity Relative dielectric constant of the particle, Surface conductivity of the particle and is given as follows Apparent resistivity of the particle Particle contact radius Coefficient of inter-particle friction Normal spring stiffness Tangential spring constant Axial stress Deviator stress Mean stress Fabric anisotropy Micro-electro-mechanical systems Discrete particle simulation Particle in cell Distance between the particles Work of Adhesion Micrometer Light Emitting Diodes Johnson, Kendall and Roberts Derjaguin, Muller and Toporov

E0 D

bp

kp

sprc Kn Kt

q p MEMS DPS PIC R WA m LED JKR DMT

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