Chapter 26 A Mechanistic Description of Granule Deformation and Breakage

CHAPTER 26 A Mechanistic Description of Granule Deformation and Breakage Yuen Sin Cheong,a,c Chirangano Mangwandi,a Jinsheng Fu,a Michael J. Adams,b ...

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CHAPTER 26

A Mechanistic Description of Granule Deformation and Breakage Yuen Sin Cheong,a,c Chirangano Mangwandi,a Jinsheng Fu,a Michael J. Adams,b Michael J. Hounslowa and Agba D. Salmana, a

Department of Chemical and Process Engineering,University of She⁄eld, Newcastle Street, She⁄eld, S13JD, UK b Centre for Formulation Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK c Presently at Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street,Cambridge,CB2 3QZ, UK

Contents 1. Introduction 2. Origin of granule strength 2.1. Autoadhesion 2.2. Wettability and surface energy of solids 2.3. Adhesion models 2.3.1. Non-deformable solids 2.3.2. Deformable solids 2.4. Friction models 2.4.1. Static friction (adhesive peeling) 2.4.2. Sliding friction 2.5. Liquid bridges 2.5.1. Static capillary force 2.5.2. Viscous junctions 2.6. Solid bridges 3. Macroscopic granule strength 3.1. Micromechanical descriptions of granule strength 3.1.1. Ensemble elastic modulus 3.1.2. Rumpf’s theory of granule strength 3.1.3. Kendall’s theory of granule strength 3.2. Measurement of granule strength 3.2.1. Diametric compression experiments 3.2.2. Impact experiments 3.2.3. Multi-granule testing 3.2.4. Quantiﬁcation of breakage propensity

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Corresponding author. Tel: +44-114-222-7560; Fax: +44-114-222-7501; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12029-7

r 2007 Elsevier B.V. All rights reserved.

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3.3. Discrete element method (DEM) simulations 1099 3.3.1. Diametric compression simulations 1099 3.3.2. Impact simulations 1100 4. Practical considerations 1102 4.1. Understanding wet granulation mechanisms 1102 4.1.1. Effect of impact velocity 1105 4.1.2. Effect of primary particle size 1106 4.1.3. Effect of binder content, binder viscosity and binder surface tension 1107 4.1.4. Remarks 1109 4.2. Targeted performance 1109 5. Concluding remarks 1113 References 1116

1. INTRODUCTION Particulate materials exist ubiquitously in both nature and industries as assemblies of discrete solid grains making intimate contact with each other where the interstitial matrix may be occupied by air or a liquid. At the gigantic scale, 10% of the earth’s surface is covered by natural particle assemblies such as sand dunes in deserts, soil sediments and pebble beaches [1]. In contrast, particle assemblies at small length scales include daily consumer products such as individual coffee or detergent granules and pharmaceutical tablets. Such particulate systems not only differ in their macroscopic size but their constituent particles also span several orders of magnitude in size. These seemingly simple systems have tremendous impact on human affairs and the global economy. In industry, the amount of particulate solids or powders being processed in the world is enormous and only second to water [2]. Approximately 10 billion metric tonnes of particulate products are manufactured worldwide every year utilising nearly 10% of the total energy produced [1]. Most of these products range from low-cost raw agricultural grains and construction aggregates to high value-added particles manufactured through sophisticated processes in the pharmaceutical and chemical industries. As pointed out by Iveson et al. [3], particulate solids account for approximately 80% of the ﬁnal and intermediate products in the chemical industries in the United States. The consequent annual revenues generated by these industries amount to US$1 trillion. In addition, the ceramics, electronics and biological sectors are driving rapidly the development of advanced particulate materials at the nanometre scale. The behaviour of particle ensembles is rich and differs considerably from that of ordinary solids, liquids and gases [4]. One may recall that the measurement of time once relied upon the ability of sand to ﬂow freely through an hourglass in a liquid-like manner. In spite of this similarity to the liquid state, sand piles up as a heap at an angle of repose due to friction, instead of spreading across a surface to form a thin layer as seen with liquids [2]. For cohesive powders such as

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cornﬂour, ﬂowability is hindered by the adhesive and frictional forces operating between the particles, unlike the free-ﬂowing sand in an hourglass. In a pack of breakfast cereal, larger nuts are usually found to accumulate at the top of a bed of smaller grains. This anomaly is called the ‘‘Brazilian nut’’ effect, which is counter-intuitive to the expectation that larger and heavier nuts should settle to the bottom of the cereal pack [5]. Despite engineers having long standing interest in the static deformation and ﬂow of particle assemblies, the underlying principles of these phenomena are still not well understood and a consensus has not been reached [2]. As a result, the design of operations and equipment for particle processing is often inefﬁcient and failure is sometimes unexpected. For instance, chemical engineers may have to deal with the blockage of silos and hoppers caused by arch formation and poor solid ﬂowability during discharge of ﬁne powders. One of the usual practices is to hammer the silo or hopper wall to encourage ﬂow but the problem is only temporarily remedied [1]. Furthermore, capillary condensation of moisture induces the formation of liquid bridges between hydrophilic particles in a humid and warm environment. Soluble materials dissolved in the contacting liquid bridges may subsequently recrystallise to form solid bonds that lead to caking of materials. Strong massive caked solids may eventually cause silos to collapse and an additional milling operation is required to crush the cake to recover the products [6]. Moreover, size segregation is inevitable when particles of different sizes are mixed mechanically, for example, in pharmaceutical industries. Consequently, preferential accumulation of materials according to particle size will occur in a similar way to the ‘‘Brazilian nut’’ phenomenon. Adversely, this results in the inhomogenous distribution of ingredients in the ﬁnal products leading to a large number of rejects. Another aspect is that ﬁne particles generated from comminution and conveying disperse in air as dust. If the particles are triboelectrically charged they may detonate causing devastation of workplace and loss of human lives. In addition, dust inhalation has an adverse health effect since some materials, which are harmless in the non-particulate solid state, become toxic after being ﬁnely comminuted [7]. In tackling the above problems, it has been proven useful to aggregate ﬁne particles to form larger entities known as granules [3]. This can be achieved through a wet granulation process where ﬁne particles are agitated with a liquid binder in a mechanical mixer. The binder may solidify upon a separate drying process to form solid bridges holding the particles together. When the addition of binder is undesirable, granulation can be facilitated using methods such as roller compaction and pressure swing ﬂuidised bed granulation [8]. These size enlargement techniques are widely used in various sectors ranging from the pharmaceutical and detergent industries to ceramic and mineral processing. Generally, pharmaceutical and detergent products are sold to the end users as granules while ceramic and mineral powders are granulated as intermediate ‘‘green bodies’’ for further processing. When ﬁne powders are agglomerated to

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form larger masses, the ratio of the interparticle attractive potentials to the gravitational force is reduced signiﬁcantly. Hence, the cohesive forces, which are dominant for ﬁne particles inhibiting particle movement, can be overcome by the granule weight resulting in improved ﬂowability. This leads to more accurate metering and easier material handling as the blockage of storage silos and conveying pipes is reduced. Ingredients that tend to segregate can be bound together to obtain product with uniform compositions as desired in pharmaceutical industries. Furthermore, the loss of precious materials and the release of hazardous substances through dust emission can be minimised with a reduced risk of dust explosion. Another advantage of size enlargement is that granules may dissolve more effectively as they tend to disperse readily compared to their constituent particles that are likely to ‘‘clump’’ when in contact with a ﬂuid. In a powder processing plant, granules are subjected to various external mechanical stresses during the different manufacturing routes such as the pharmaceutical production scheme depicted schematically in Fig. 1. Granule strength is the key issue in determining the efﬁciency of the processes highlighted in grey. For example, collisions between granules or the impact of granules with the mixer wall and impeller are observed in high shear mixers causing attrition and breakage of the granules [3,10]. There is evidence suggesting that granule breakage is useful in promoting a more uniform distribution of the binder within a given batch of granules [11]. After high shear granulation, the granules may be discharged onto a vibratory conveyor belt and transported to a storage silo. During storage, granules in the lower part of the silo experience substantial static compressive forces that arise from the weight of the granules accumulating above. Hence, once granules are formed, they should be sufﬁciently strong to resist these handling and storage damages to avoid product degradation and debris generation.

Crystalliser

Liquid formulation & filling lines

Filter

Blending

Dissolution

Dryer

High shear granulation

Tabletting

Mill

Granule conveying

Storage

Fig. 1. Flowchart showing typical granule processing routes (highlighted in grey) in pharmaceutical industries. (After Barrett et al. [9], with permission.)

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Some drugs in the pharmaceutical industries are designed to be administered directly to the human body through inhalation [12–14]. This is useful in delivering drugs that may cause adverse side effects with the appropriate dosage to the target site only and is commonly used in pulmonary disease treatment. Hence, it is desirable to manufacture these drugs in a granular state, which can be disintegrated and dispersed readily to form an aerosol while having sufﬁcient strength to survive the mechanical stresses during handling and transportation. Moreover, for example, detergent granules manufactured in the form of end products should be sufﬁciently weak to disintegrate and disperse readily for rapid dissolution [15]. It is plausible to improve the dissolution properties by making granules with a high air volume fraction or porosity but to the detriment of the strength. In contrast, it is essential to achieve complete crushing of granules in a compaction process to ensure that tablets with a homogenous density distribution and appropriate strength are produced. The presence of strong granules as inclusions may act as defects in the tablets. Consequently, cracks are likely to be initiated from these ﬂaw sites causing the tablets to fracture. Similarly, a substantial reduction in the strength of ceramic structures is expected if granules, which act as the intermediate green bodies, are not eliminated completely during the forming process [16]. Thus, granule strength should be tailored carefully to seek the balance between the resistance to handling damages and end-use requirements. Typically, the minimum strength of a granule is expected to range from 1 to 10 MPa based on the tensile strength measured for dry compacted pharmaceutical granules handled in actual processing plants [17]. Granule strength is also important in granulation processes. The binder is typically a solution or a melt and causes the feed particles to granulate due to the formation of capillary and viscous liquid bridges [3]. If the granules are weak they will be broken down by the mechanical action of the granulator but, if they are too strong, very large granules will be formed. Thus, it is important to control the strength of the forming granules to achieve a product that has the required granule size distribution. This chapter attempts to reiterate that granule strength stems from the interparticle adhesion and friction forces, which is central to the control of granule strength; the term adhesion is used broadly here to include any type of interaction that binds the primary particles together including solid bonds caused by the presence of a binder. Therefore, the engineering of granule strength can be achieved qualitatively by manipulating the interfacial properties of particle surfaces and the interfacial and bulk properties of the binder, if present. This approach is of great importance in pharmaceutical industries that has stringent time limits to release a product to the market. Furthermore, strict legislation and ﬁnancial constraints make trial and error experiments, which require large amounts of pharmaceutical materials, impractical. Further demand for this correlation is clear after the launch of the good manufacturing practice (GMP) and process

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analytical technology (PAT) initiatives. These initiatives require the food and drug industries to implement control strategies on their manufacturing processes, aiming for effective and efﬁcient production with reduced wastage of materials [18]. The successful prediction of granule strength from the microscopic particle interactions is believed to allow these industries to incorporate appropriate process control to fulﬁll the objectives of the GMP and PAT initiatives.

2. ORIGIN OF GRANULE STRENGTH As mentioned in the Introduction, various size enlargement processes, either wet or dry, with different degrees of agitation are available to produce granules with complex packing structures (distribution of contacts) and bonding mechanisms. Despite this, granules can be treated as an assembly of discrete particles where external perturbations are transmitted via the interparticle contacts in the normal and tangential directions. This section contains a brief account of existing theories and techniques employed to describe the interparticle interactions including autoadhesion, friction, liquid bridges and solid bonding. These microscopic interactions control the macroscopic elasticity and strength of granules based on the packing of the constituent particles, as will be described in Section 3.1.

2.1. Autoadhesion Despite the existence of molecular adhesion, its effect is not apparent if the separation between the surfaces of two distinct solids is large compared to the equilibrium separation between atoms, z0 (typically a few angstroms). The parameter z0 is the distance at which the attractive and repulsive forces between neighbouring atoms equilibrate. It has been observed that two approaching solid surfaces jump into contact when the separation is reduced to a dimension close to z0 [19]. This is due to the fact that the surfaces are now within the range of action for molecular attractive forces to operate thus, pulling the surfaces together. This phenomenon is more profound if the surfaces are clean and smooth. Furthermore, the attractive forces become comparable to the weight of the contacting solids, as the body size is made smaller. The adhesion force scales with the ﬁrst power of the solid size as discussed in Section 2.3. Comparing the relative magnitudes of adhesion and gravity forces acting on a solid while reducing the body size, the gravity force (weight) diminishes more rapidly than the adhesion force, since it depends on the third power of size [20]. Following the above arguments, it is not surprising that molecular adhesion is the main mechanism responsible for the structural integrity of a binderless granule composed of ﬁne constituent particles. During a granulation process, the

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constituent particles are compacted by mechanical agitation to form granules with a close-packed structure. As a result, the separations between the particles are reduced to within the range of action for molecular adhesion to operate. The adhesion is further enhanced if the particles are of sub-micrometer size. However, it should be noted that the surfaces of real particles are seldom smooth. Hence, the real contact between the constituent particles within a single granule is commonly due to the pairwise contacts between asperities of the particles. Molecular adhesion can be interpreted in terms of energy since work is required to pull apart the contacting solid surfaces. As illustrated in Fig. 2, an interface between the solid surfaces is formed when contact is established. The energy corresponding to the formation of a unit area of this contact is called the interface energy, denoted g12. An attempt to separate the contacting surfaces involves not only the elimination of the solid–solid interface but also the creation of two new solid surfaces. The amount of reversible work required to separate a unit area of the surfaces from contact to inﬁnity is deﬁned as the work of adhesion, WA, which can be written as the Dupre´ equation shown below [21]. W A ¼ g1 þ g2 g12

ð1Þ

where g1 and g2 are the surface energies of two dissimilar solids 1 and 2, respectively. The surface energy of a solid is the energy expended to form a unit area of the solid. For normal ambient conditions, adsorption of foreign vapour onto the solid surface is inevitable. Hence, the solid surface energy measured in air is much smaller than that obtained in vacuum. For two identical solids in contact, each with a surface energy of gS in vacuum, equation (1) reduces to W A ¼ 2gS

ð2Þ

since g1 ¼ g2 ¼ gS and g12E0 [22]. The determination of the work of adhesion according to equations (1) and (2) is difﬁcult by direct measurement except for a few special cases when atomically smooth interfaces may be formed, e.g. mica and some elastomers. However, a reasonable estimate of these energy

Solid 1, 1 Solid 1, 1 CONTACT

−12

SEPARATION Interface, 12

WA

Solid 2, 2 Solid 2, 2

Fig. 2. Schematic diagram illustrating the total energy change (work of adhesion, WA) associated with the separation of a contact interface.

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parameters can be made based on the wetting behaviour of the solid [23], as described later.

2.2. Wettability and surface energy of solids The wettability of a solid refers to the tendency of a liquid drop to spread on its surface. For a given solid surface, only limited spreading will occur for a drop of a partially wetting liquid so that a ﬁnite contact angle, yE, is formed between the liquid–vapour interface and the solid surface, as illustrated in Fig. 3. At equilibrium, the energy of this solid–liquid–vapour system can be described using the well-established Young’s equation (see [24]) gSV ¼ gLV cos yE þ gSL

ð3Þ

where gSL is the solid–liquid interface energy while gSV and gLV are the solid surface energy and liquid surface tension, respectively. There is a reduction in the solid surface energy caused by the adsorption of vapour on a solid surface. When this is taken into account by including the spreading pressure, pe ¼ gSgSV equation (3) becomes [25] gS ¼ gLV cos yE þ gSL þ pe

ð4Þ

gS being the surface energy of solid in vacuum. It appears from equation (4) that the surface energy of a solid, which determines the work of adhesion between two contacting solids, can be measured from contact angle experiments. Unfortunately, these experiments suffer from a drawback that gSL and pe are not known a priori and independent measurements of these parameters are not reliable. Instead of measuring all the variables in the Young’s equation, Fox and Zisman [26] suggested that gS for a given solid can be estimated from the contact angles of a homologous series of liquids on the same solid surface. They observed that a straight line can be ﬁtted through a Fox–Zisman plot of cosyE as a function of gLV for the series of liquids. A measure of the solid surface energy is given by the critical surface tension gC obtained by extrapolating the straight line to cosyE ¼ 1. This parameter signiﬁes that liquids with a surface tension that is less than gC spread instantaneously over the solid surface. By inspection of equation (4), the Liquid drop

E Solid surface

Fig. 3. A typical drop proﬁle of a non-wetting liquid resting on a solid surface.

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implications of this empirical approach are that pe is negligible and gSL vanishes as cosyE approaches unity. Despite the lack of evidence for these assumptions, this method has been proven to provide satisfactory estimates of the surface energy of low-energy solids such as hydrocarbon surfaces with certain pure liquids interacting via dispersion forces only [27]. For other systems where simultaneous dispersion and non-dispersion forces (polar interactions such as hydrogen bonding) exist, the total surface energy of a single phase gT can be decomposed into energies due to dispersive gD and nondispersive gP interactions [28]. gT ¼ gD þ g P

ð5Þ

According to Wu [29], the total solid–liquid interface energy for a system comprising both dispersive and non-dispersive interactions can be described by gTSL ¼ gTS þ gTLV

D 4gD 4gPS gPLV S gLV D gD gPS þ gPLV S þ gLV

ð6Þ

Thus, substitution of equation (6) into equation (4) and assuming pe 0 gives gTLV ð1 þ cos yE Þ ¼

D 4gD 4gPS gPLV S gLV þ D gD gPS þ gPLV S þ gLV

ð7Þ

Since the dispersive and non-dispersive components for the surface tensions are known, it can be seen from equation (7) that the total surface energy of a solid can be estimated by measuring the contact angle of two non-wetting liquids on the solid surface. Such an investigation was undertaken by El-Shimi and Goddard [27] where methylene iodide with mainly dispersion forces and water having both dispersion and non-dispersion forces were used as the probe liquids on lowenergy hydrocarbon surfaces and human skin. Close agreement was found between the solid surface energies calculated from equation (7) and the critical surface tensions extracted using the Fox–Zisman method, which was shown to be applicable for low-energy surfaces.

2.3. Adhesion models The work of adhesion can be inferred by considering the total force required to separate two contacting surfaces, which is known as the pull-off force. As reviewed by Tabor [22], the theoretical calculations of the pull-o¡ force can be performed based on the principles set forth in the proceeding sections. It is essential to note that the contacting solids are treated as continua with molecularly smooth surfaces in these theoretical considerations. In relation to particulate systems, discrete particles are approximated as solid spheres in these theories. However, the results are still applicable for rough grains since the

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contact geometry between a pair of asperities approximately coincides with that of two spherical surfaces, i.e. curved surfaces each deﬁned by a radius of curvature [30]. The adhesion effect arising from surface attraction is commonly known as autoadhesion.

2.3.1. Non-deformable solids The mutual attractive forces between individual atoms at sufﬁcient distances apart are mainly electrostatic in nature. These attractive forces are commonly referred to as dispersion forces and were shown to be proportional to the inverse seventh power of the distance between the two atoms by London (see [21]). Using pairwise addition of the dispersion forces between all the atoms in two non-deformable (hard and rigid) solid spheres, Bradley [31] derived the following expression for the pull-off force, PC, exerted between macroscopic spheres: P C ¼ 2pR W A

ð8Þ

with the effective radius, R ¼ (R1R2)/(R1+R2) where R1 and R2 are the radii of the contacting spheres; note that R is equal to the radius of a sphere in contact with a ﬂat surface. Experimentally, for example, the value of PC required to separate silica spheres of different sizes has been found to vary with the effective radius as a straight line passing through the origin [31]. The ﬁtting of equation (8) to the experimental data yielded a work of adhesion of 33.8 mJ m2. This value is smaller than that of 50 mJ m2 measured by Kendall et al. [32] but comparable to the value of 36 mJ m2 obtained by Heim et al. [33]. The discrepancy in different experiments may be attributed to the difference in the surface conditions of the silica spheres, such as the roughness and contamination. The rigid sphere assumption of Bradley [31] implies that the solids were in point contact with no distortion of the contact region. Real elastic spheres tend to ﬂatten under the action of the surface attractive forces resulting in a circular contact area. This phenomenon was observed experimentally when rubber spheres were placed in contact under small or no externally applied force [19]. Hence, the elastic deformation of solids must be accounted for to provide a more realistic model for the adhesion between solid particles.

2.3.2. Deformable solids The development of the adhesion models describing the effect of surface adhesion forces on the elastic deformation of solid particles was based on the well-established Hertz theory [34]. Therefore, it is necessary to summarise this theory for two contacting elastic spheres loaded externally in the absence of surface forces.

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2.3.2.1. Hertz contact model The contact problem between two smooth and frictionless elastic spheres was studied by Hertz [34] where the deformations of the spheres were restricted to the contact region as justiﬁed by the high local stresses involved. According to Hertz [34], the radius of the contact circle, a, formed when two such spheres are pressed together, varies with an externally applied force of, P, as follows: a3 ¼

3R P 4E

ð9Þ

where 1 n21 1 n22 1 ¼ þ E E1 E2

ð10Þ

such that E1, E2, n1 and n2 represent the Young’s moduli and Poisson’s ratios of the spheres, respectively. The distribution of the compressive pressure within the contact is such that it attains a maximum at the centre of the contact circle and decays to zero at the periphery as sketched in Fig. 4. According to this deformed proﬁle, the displacement of the centres of the spheres towards each other, d, given by d¼

a2 R

ð11Þ

It follows from equations (9) and (11) that the load–displacement relationship takes the following non-linear form: P¼

4 1=2 3=2 E R d 3

ð12Þ

The Hertz analysis is a good approximation for large bodies when the externally applied force is large compared to the surface forces. Nevertheless, it is Hertzian contact pressure distribution

H (r, )

z0 a r

Fig. 4. The Hertzian contact proﬁles of two elastic spheres in contact (adapted from Tabor [22]). Note that the equilibrium separation z0 is exaggerated for clarity.

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commonly observed that the contact area is larger than the Hertz prediction as the external force approaches zero. This is a consequence of the dominant contribution from the surface forces to the elastic deformation of the spheres at low external forces. Hence, the elastic contact deformation of lightly loaded solid spheres can be related to the work of adhesion, WA required to pull the spheres apart. The determination of WA is dependent on the exact shape of the deformed surface near the contact region. There are two main results based on two different assumed deformation proﬁles.

2.3.2.2. Derjaguin, Muller and Toporov (DMT) adhesion model In the study of Derjaguin and co-workers [35], the deformed surfaces outside the contact of two adhering spheres were postulated to follow the Hertzian proﬁle as ﬁrst proposed by Derjaguin [36], even though the surface attraction cause a larger contact area than the Hertz prediction. In principle, the elastic reaction force of the deformed spheres is balanced by the surface attractive forces and any externally applied force. The total surface force not only consists of the contact forces within the enlarged contact area but also the noncontact forces exerted in the annulus region surrounding the contact. Thus, the total surface adhesion energy WS0 can be written as the sum of the molecular energies Wcontact and Wnon-contact due to the contact and non-contact forces deﬁned above. W 0S ¼ W contact þ W non-contact

ð13Þ

To calculate the total surface force pulling the spheres together, one can consider the rate of change of W 0S by varying the displacement between the centres of the spheres, d. Separation of the spheres occurs as d-0 and hence the pulloff force PC corresponds to the PS at point contact dW 0S PC ¼ ¼ 2pR W A ð14Þ dd d!0 which resembles the result of Bradley [31] in equation (8), even after taking elastic deformation into account. Perhaps, this result is not surprising as Derjaguin et al. [35] emphasised that the DMT analysis was intended for small and hard particles with large elastic moduli.

2.3.2.3. Johnson, Kendall and Roberts (JKR) adhesion model In contrast to the DMT analysis, Johnson et al. [19] suggested that the stress distribution within the contact area must be altered from the Hertzian distribution by the surface attraction and that all the attractive forces act within the contact area. The contact stress distribution can be found by the superposition of the compressive Hertzian distribution for an enlarged contact area and a

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JKR contact pressure distribution

z0 a

Fig. 5. Figure showing the JKR contact proﬁles for two elastic spheres in contact in the presence of surface attraction (after Johnson et al. [19]).

tensile stress distribution similar to that generated by a rigid punch pressing onto an elastic plane [37]. This hypothesis gives rise to an inﬁnite tensile stress at the boundary of the contact circle causing the contact surfaces to meet perpendicularly at just outside the contact area forming a neck (see Fig. 5). However, slight peeling of the sphere surfaces near the edge of the contact area must occur in reality such that the surface attraction is maintained in equilibrium by a ﬁnite tensile stress as argued by Johnson [37]. Consider two solid spheres brought into contact from inﬁnity by a small external force, a surface adhesion energy, WS, is released due to the formation of the enlarged circular contact interface. To ensure equilibrium, the total energy of the system, WT, must achieve a minimum where the elastic energy stored in the deformed spheres, WE, balances the expended potential energy of the external force, WM, and that dissipated in surface adhesion energy, WS. Therefore, WT ¼ WE WM WS

ð15Þ

On this basis, the expression relating the enlarged contact radius, a, to the external force, P, is given by [19] qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3R a3 ¼ P þ 3pR W þ 6pR W A P þ ð3pR W A Þ2 ð16Þ A 4E It can be seen that this expression reduces to the Hertz equation (equation (9)) if surface attractions are neglected for which WA ¼ 0. The total normal force acting over the contact area, PJKR is qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð17Þ P JKR ¼ P þ 3pR W A þ 6pR W A P þ ð3pR W A Þ2

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If an external tensile force (i.e. negative P) is applied to separate the spheres from contact to inﬁnity, the pull-off force PC expressed below is obtained such that a real solution to equation (16) exists [19] P C ¼ 1:5pR W A

ð18Þ

t is worth noting that the JKR pull-off force is 25% smaller in magnitude than that of the DMT analysis (equation (14)) as a result of the modiﬁed contact stress distribution. Furthermore, detachment occurs at a ﬁnite contact radius, unlike the gradual diminution of the contact area to a point contact predicted by the DMT. If the peeling process of the contact interface is viewed as a Mode I fracture (crack opening), there is an analogy to Grifﬁth’s criterion [38] for the fracture of brittle solids to describe the detachment of the spheres. Hence, the JKR pull-off condition corresponding to the sudden separation of the spheres in a ‘‘jump-wise’’ manner is expected once the rate of release of the stored elastic energy in the contact exceeds the rate of gain in the surface adhesion energy (see [39]). To verify the JKR analysis, smooth and soft elastomer spheres were used since it was argued that dust particles or surface asperities may be pushed into the bulk of the elastomer due to the small elastic modulus, thus permitting good intimate contact [19]. The variation of the contact radius of the elastomer spheres with the applied force was found to agree remarkably well with equation (16) when the work of adhesion was taken to be 71 mJ m2. This implied that the surface energy of each elastomer surface was approximately 35 mJ m2 according to equation (2). The work of adhesion was reduced by an order of magnitude when the elastomer contact was wetted with water. These energy values were proved to be convincing when inserted into Young’s equation resulting in a contact angle of 641 for water on the elastomer contact. This value was consistent with the measured value of 661 for a water drop resting on the same elastomer surface. Furthermore, the JKR theory was supported by experiments where soft gelatine spheres of different sizes were pressed onto Perspex ﬂat [19]. When a tensile force is applied to separate elastomer spheres, it has been observed that large uncertainty in the contact radius measurement was inevitable near the pull-off condition as the attainment of contact equilibrium was prolonged due to the viscoelastic behaviour of elastomers [19]. In spite of this, the dependency of the contact radius on the applied force agreed reasonably with the JKR theory within the limits of experimental uncertainties. In addition, the contact interface was observed to detach in an unstable way. Apart from the original experimental veriﬁcation, the JKR theory was realized experimentally for adhesive contact between a wide range of materials in different contact geometries. For example, these include the crossed cylinder conﬁgurations of the polymer ﬁbre monoﬁlaments studied by Briscoe and Kremnitzer [40]

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and the mica on mica contact investigated by Homola et al. [41] using the surface force apparatus (SFA). Recently, the contact between a 140 nm cantilever tip and a mica surface was shown to exhibit JKR behaviour in the atomic force microscopy (AFM) study of Carpick et al. [42]. The theory was shown to break down when surface roughnesses comparable to atomic diameter were considered in the molecular dynamic simulation of Luan and Robbins [43].

2.3.2.4. Limits of DMT and JKR Despite different deformation proﬁles being assumed, the DMT and JKR analyses were demonstrated to yield the same dependency of the pull-off force on the particle radius [22]. Thus, it is difﬁcult to ascertain the selection of the appropriate model to describe the adhesion of elastic particles, apart from using nature of the detachment of the particles as a criterion. A formal approach follows the suggestion of Tabor [22] to identify the limits of application of the DMT and JKR theories using the following dimensionless parameter !1=3 R W 2A z¼ ð19Þ E 2 z30 Essentially, this parameter may be interpreted as the ratio of the neck formed around the contact circle to the range of action of the surface forces characterised by the equilibrium separation, z0. As Tabor [22] pointed out, the neck for two large adhering particles with small elastic moduli like elastomers is large compared to z0. In this regard, the surface forces outside the contact area diminishes rapid with separation between surfaces and can be neglected in accord with the JKR theory. In the other extreme, the DMT is applicable to describe the adhesion between small and hard particles where the neck size is small to permit the action of surface forces outside the contact area. A form of dimensionless group similar to z but with a prefactor of 2.92 was derived by Muller et al. [44]. It was shown that the pull-off force resembles the DMT prediction for zo1, whereas a transition to the JKR pull-off occurs at z45. This transition is supported by the numerical computation performed by Greenwood [45].

2.4. Friction models The adhesion of solid particles summarised in the preceding section represents only the normal interaction between solid surfaces. When a contact interface is subjected to tangential loading, relative motion is resisted by a friction force operating in the direction opposing the tangential force. This action of the friction force is dissipative in nature and may be regarded as the work per

Y.S. Cheong et al.

1070

unit sliding distance. The description of the friction force, F, acting between two non-adhering macroscopic bodies is based on the Amonton’s law m¼

F P

ð20Þ

where P is the external force applied normally to the system and m the coefﬁcient of friction. Sliding between two surfaces is not possible unless a limiting value of the frictional force is reached, i.e. FomsP, where ms is the coefﬁcient of static friction. Once sliding occurs, the relationship in equation (20) still applies to describe sliding friction although the value of the dynamic value of m may be different from the static value. Numerous observations have shown that the frictional force is independent of the geometric area of contact and the velocity at which two contacting bodies slide over each other. Nevertheless, it is clear that equation (20) is not applicable for all cases in practice since friction still exists when the external load is reduced to zero or even made negative, especially in the context of cohesive particles (see, for example, Gao et al. [46]). In describing the frictional contacts of cohesive powders, the commonly used approach is Coulomb’s law in the following form: F P ¼ tc þ m A A

ð21Þ

where A is the cross-sectional area of the powder assembly under consideration and tc is the cohesive shear strength of the powder while other symbols are as deﬁned in equation (20). It can be seen that friction is now ﬁnite at zero applied force due to the contribution from the attractive forces operating between ﬁne particles characterised by the parameter tc. However, it was reported by Kendall [47] that the frictional force for ﬁne powders is incompatible with Coulomb’s law where a signiﬁcant reduction of friction force was noticed at high applied forces. Furthermore, smaller particles are usually associated with friction coefﬁcients larger than that measured for macroscopic blocks of the same material [47]. According to the argument of Kendall [47], the inadequacy of the Coulomb’s law is attributed to the assumption that the externally applied force and the surface attraction between ﬁne particles are independent of each other. Thus, it is essential to consider the interdependence between friction and adhesion. A special feature of particles is that they may form point contacts even when their surfaces are rough. This arises because contact may occur at single asperities due to the acute radii of particles. The friction of most particles corresponds to the adhesion mechanism in which temporary junctions, formed by short-range attractive forces such as van der Waals interactions, are intermittently ruptured [48]. Such an interfacial process is common because most particles are relatively stiff and elastic so that bulk dissipation processes are negligible. The interfacial frictional force is related to the real area of contact, AR,

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by the following relationship: F ¼ ti A R

ð22Þ

where ti is the interfacial shear stress required to rupture the adhesive junctions. It should be noted that this does not imply that there is an adhesive force between the particles at a zero applied load since the real area of contact is then also zero according to this relationship. Since the real area of contact increases as the 2/3 power of the normal load according to the Hertz equation (9), the frictional force will also vary in this way provided that the surfaces of the particles are clean. In practice, most particles are contaminated with organic materials that exhibit a pressure dependence of the interfacial shear strength so that the load index is in the range of 2/3 to unity [49].

2.4.1. Static friction (adhesive peeling) The effect of a tangential force, T (i.e. equal and opposite to friction force) on the behaviour of an adhesive contact between two elastic spheres pressed together by an external force was investigated by Savkoor and Briggs [39] by extending the JKR adhesion model. According to their theoretical formulation, the contact radius a between the spheres can be related to the external normal force P and the friction force F as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2 3R 4 T 2E5 a3 ¼ ð23Þ 6pW A P þ 9W 2A p2 R2 P þ 3pR W A þ 4E 4G where 1 2 n1 2 n2 þ ¼ G G1 G2

ð24Þ

Comparing equations (16) and (23) with the normal force kept constant, it can be seen that the contact radius diminishes with increasing tangential force leading to a ‘‘peeling’’ mechanism. This ‘‘peeling’’ process proceeds in a stable manner and will be complete once T reaches a critical value, TC, thus rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ G T C ¼ 2 ð6pW A P þ 9W 2A p2 R2 Þ ð25Þ E and the contact radius diminishes to a3C ¼

3 R ðP þ 3pR W A Þ 4 E

ð26Þ

The contact radius measured in the experiments of Savkoor and Briggs [39], where a rubber hemisphere was pressed onto a glass slide at different normal loads, exhibited clear reduction under a monotonically increasing tangential

1072

Y.S. Cheong et al.

force. In addition, it was concluded that the data were in reasonable agreement with equation (23) within the limits of experimental error. The different processes that may occur at the end of contact peeling will be considered in the following subsection.

2.4.2. Sliding friction In describing the sliding friction between small cohesive particles, it was pointed out by Kendall [47] that the external force and the surface attraction is interrelated such that the effective load acting over the contact interface PJKR is given by equation (17). Consequently, a modiﬁed form of Coulomb’s law is obtained based on Amonton’s law as follows: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ T ¼ m P þ 3pR W A þ 6pR W A P þ ð3pR W A Þ2 ð27Þ Qualitatively, this result supports the dependence of friction on particle size observed for ﬁne powders and the anomaly where friction decreases at high applied force as indicated by the variation in the gradient of equation (27). In the sliding experiments carried out by Briscoe and Kremnitzer [40] using adhesive polyethylene–terephthalate monoﬁlaments, the measured tangential force as a function of normal applied force was found to follow a non-linear trend prescribed by equation (27). Despite this, ﬁtting of the theory to the experimental data required a work of adhesion, which was inconsistent with the measured surface energy of the polymer [47]. Presumably, this can be explained by the failure of the model to account for contact peeling prior to the onset of sliding. Alternatively, the data may be explained by a power law relationship as mentioned previously such that the normal load is the sum of the applied and adhesive values according to DMT theory [50]. Experimental evidence for a reduction in the contact size emerged from the recent AFM study of Carpick et al. [42] in accordance with the analysis of Johnson [30]. A more rigorous analysis was performed by Thornton [51] to examine the conditions at the end of the contact peeling (TZTC), in which the contact interface between the adhering elastic spheres was ﬁrst assumed to peel followed by a smooth transition to complete sliding. This assumption coincides with the conclusion of Johnson [30] by treating the contact interface, in the presence of normal and tangential forces, as a crack using the principles of fracture mechanics. Recalling the resultant normal stress distribution for the JKR adhesion model, there is a central portion of the contact area of radius a1 which is subjected to compressive stresses. This compressive region is surrounded by an annulus of tensile stresses rising to an inﬁnite value at the contact periphery (see [52]). It was argued that peeling of the contact area was only necessary until the contact

Mechanistic Description of Granule Deformation and Breakage

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radius is reduced to [51] a31 ¼

3R P1 4E

ð28Þ

where P 1 ¼ P JKR

P JKR P 1 3P JKR

3=2 ð29Þ

with PJKR given by equation (17) and hence, T ¼ mP 1

ð30Þ

Expression (30) was found to follow the whole range of the data obtained in the sliding experiments of Briscoe and Kremnitzer [40] satisfactorily including the results at negative normal loads. It should be noted that the work of adhesion measured experimentally was used in the analyses of Thornton [51] in contrast to the inconsistent values assumed in the work of Kendall [47]. Hence, it is reasonable to accept equation (30) in preference to equation (27) based on the comparison between the two models with the experiments of Briscoe and Kremnitzer [40], although equally close agreement is obtained with the power law model discussed above.

2.5. Liquid bridges When a powder mass is stored in humid environment, a common phenomenon is capillary condensation of water vapour causing caking due to the formation of liquid bridges at the interparticle contacts. During wet granulation, however, liquid binder is deliberately added to a powder mass to induce size enlargement. The resulting wet granules can exist in several different levels of liquid saturation. The granules are in the pendular state at low liquid saturation where the primary particles are held together by discrete liquid bridges with the interstitial space between the particles ﬁlled by air. Further increases in liquid saturation caused by either continuous binder addition or granule consolidation leads to the capillary state, where the entire interstitial space is occupied by the binder (see [3]). An intermediate state between these saturation levels is known as the funicular state. In these granular states, there are additional contributions to the interparticle forces from the static capillary action and dynamic viscous effect of the liquid binder. This section will focus on the theoretical description of the forces generated due to pendular liquid bridges connecting a pair of spherical bodies only. The theoretical analyses of the funicular and capillary states are considerably more complex and 2D results are given by Urso et al. [53].

Y.S. Cheong et al.

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2.5.1. Static capillary force For a pair of stationary spheres with a small separation distance, the formation of a pendular liquid bridge connecting the spheres requires that the Gibbs free energy of the system attains a minimum value. Under such circumstances, the surface proﬁle of the bridge at a given solid–liquid contact angle has a mean curvature, xm, which can be deﬁned by two principle radii of curvature in orthogonal directions r1 and r2, respectively. Conventionally, a positive radius of curvature is drawn with its centre lying in the liquid mass. The mean curvature of the bridge surface must remain constant so that the surface tension of the liquid–vapour interface is balanced by the hydrostatic pressure difference between the interior and exterior of the liquid mass DP, as observed by Plateau [54]. This relationship is described by the Laplace–Young equation as follows: 1 1 DP ¼ gLV xm ¼ gLV þ ð31Þ r1 r2 At equilibrium, the forces at any point along the axial direction of a liquid bridge are equal otherwise there would be ﬂow of the liquid. For a gravity-free analysis, it follows that the liquid bridge force between the spheres can be computed at any axial position of the bridge by summing the contributions from the axial component of surface tension and the axial force due to the hydrostatic pressure difference, DP. For small liquid bridges, the distortion of the bridge proﬁle due to gravity can be neglected [55]. When considering a large pendular bridge with its axial direction oriented along the gravity ﬁeld, the buoyancy acting on the portions of the spheres submerged in the liquid should be accounted for in estimating the bridge force. In the context of wet granular materials, the constituent particles may be treated as rigid bodies so that the particle geometry is undistorted under the action of the liquid bridge forces. The hydrostatic pressure ﬁeld across a cross-section of a liquid bridge can be evaluated analytically by solving equation (31) numerically as demonstrated, for

rA

rC R

rN

ϕ β

A

2S

Fig. 6. The geometry of the toroidal approximation with particles of radius R, separated by a distance 2S, connected by a liquid bridge of half-ﬁlling angle b and contact angle j [55].

Mechanistic Description of Granule Deformation and Breakage

1075

example, by Erle et al. [56] and Orr et al. [57]. Nevertheless, such procedures are computationally inefﬁcient and impractical for applications such as granular dynamics computer simulations. To simplify the problem, one can adopt a toroidal approximation ﬁrst proposed by Haines [58], in which the meridional bridge proﬁle is taken as a circular arc as shown in Fig. 6. Following geometric arguments, it can be shown that [59] ðr N r A Þ ¼

A þ R sin b tanðp=2 b jÞ

ð32Þ

n o1=2 r A ¼ ð1Þn A2 þ ½R sin b ðr N r A Þ2

ð33Þ

r C ¼ R sin b

ð34Þ

n¼

0; ðb þ jÞ4p=2

1; ðb þ jÞop=2

ð35Þ

where A ¼ S+R(1cosb) such that b is the half-ﬁlling angle. In practice, one may express the half-ﬁlling angle in terms of the volume of the liquid bridge since the amount of liquid added to a powder mass can be measured reliably (see [60,61]). It may be seen that the local curvature of the toroidal proﬁle varies from ð1=r A þ 1=r N Þ1 at the neck to ð1=r A þ 1=r C Þ1 at the solid–liquid interface reﬂecting that the mean bridge curvature is not constant and hence errors are introduced in the estimated bridge forces. The accuracy of this approximation will be discussed later. Neglecting the effects of gravity, Fisher [62] proposed the gorge method to estimate the total attractive force Pgorge acting on a pair of equal spheres at the neck of a toroidal liquid bridge (see Fig. 6) P gorge ¼ 2pr N gLV pr 2N DP

ð36Þ

On the right hand side of equation (36), the ﬁrst term is the positive (attractive) contribution from the axial component of surface tension whereas the second term is the hydrostatic component, which can be positive for DPo0 depending on the bridge geometry. However, Adams and Perchard [63] argued that it is more physically appropriate to employ the boundary method to calculate the attractive force at the solid–liquid interface where forces are transmitted to the spheres, so equation (36) was modiﬁed accordingly. Both methods predict that there is a rapid decrease in the attractive bridge force with increasing separation between the spheres as indicated by the dimensionless plots in Fig. 7. Furthermore, the bridge force attains a maximum at an intermediate half-ﬁlling angle and this angle increases with the separation. The exceptional monotonic decline of the bridge force with increasing half-ﬁlling angle at zero separation is a consequence of the large pressure deﬁcit and a similar explanation applies as b

Y.S. Cheong et al.

1076 1.0

5° 10°

β = 1°

0.8

20°

0.6 F*

40°

0.4

0.2 'Gorge' 'Boundary' 0.0 10-6

10-5

10-4

(a)

10-3

10-2

10-1

100

101

S* 1.0

S* = 0

'Gorge' 'Boundary'

S * = 0.001

0.8

S * = 0.01

F*

0.6

S * = 0.1 0.4

0.2

0.0 0 (b)

10

20

30 β (°)

40

50

60

Fig. 7. The variation in the dimensionless attractive force (F ¼ F/2pRgLV) between equal spheres as a function of (a) the dimensionless half-separation distance (S ¼ S/R) with constant half-ﬁlling angle and (b) the half-ﬁlling angle with constant dimensionless halfseparation distance, evaluated using the toroidal approximation for zero contact angle by both ‘gorge’ and ‘boundary’ methods.

Mechanistic Description of Granule Deformation and Breakage

1077

diminishes (see Fig. 7(b)). Clearly, the ‘‘boundary method’’ yields a larger bridge force than that calculated using the ‘‘gorge method’’ and the predictions deviate from each other as b increases. In the theoretical work of Lian et al. [64], the gorge method was found to provide a more accurate total bridge force than the boundary method, which was within 10% of that given by exact numerical solution of the Laplace–Young equation. They further demonstrated that the accuracy of the toroidal approximation could be improved by introducing simple scaling coefﬁcients. Recently, another closeform approximation for the total bridge force as a function of a scaled dimensionless half-separation distance was developed by Willett et al. [55]. Several criteria are available to specify the critical separation distance at which a pendular bridge ruptures. Erle et al. [56] pointed out that bridge rupture is expected at a ﬁnite rather than a zero neck radius. According to the hypothesis of De Bisschop and Rigole [65], bridge rupture occurs when the half-ﬁlling angle is reduced to a critical value. However, this hypothesis was shown to be incorrect by Mazzone et al. [66], who identiﬁed that the bridge ruptures at a critical separation distance beyond which no stable liquid bridge can be formed since no solution to the Laplace–Young equation exists. On this basis, Lian et al. [64] found that this critical separation distance is given by the following empirical expression: 2Sc ¼ ð1 þ 0:5jÞV 1=3

ð37Þ

One of the most comprehensive experimental measurements of the forceseparation curves for liquid bridges were performed by Mason and Clark [67] using 30 mm polythene hemispheres connected by liquid bridges of different volumes. The gravitational distortion of the bridge proﬁle and buoyancy effect were minimised by immersing the hemispheres in water, which has the same density as the bridge liquid (a mixture of di-n-butyl phthalate and liquid parafﬁn). The force-separation curves resulted from numerical solution of the Laplace–Young equation and the gorge toroidal approximation were shown to exhibit comparable agreement with most experimental data of Mason and Clark (see [64]). However, deviation of the calculated bridge forces from the measured forces was observed at small separation and large bridge volume. Similar discrepancy at small separation, where the measured forces were lower than expected, was observed by Erle et al. [56] and Mazzone et al. [66]. However, this phenomenon was not observed in the experimental studies of Willett et al. [55] with unequally sized sapphire spheres having a zero solid–liquid contact angle. Moreover, the rupture distances measured by Mason and Clark were observed to follow equation (37) reasonably well as shown by Lian et al. [64]. Fairbrother and Simons [68] quoted a 10% deviation of the measured rupture distances from the predictions of equation (37) for equal spheres of glass ballotini connected by liquids having various solid–liquid contact angles.

1078

Y.S. Cheong et al.

For large liquid volumes and large spheres, the effect of gravity can no longer be neglected and this was considered in detail by Mazzone et al. [66] and Adams et al. [69]. Essentially, the liquid proﬁle bridging a pair of equal spheres with the axial direction oriented along the gravity ﬁeld (rotating Fig. 6 by 901) was shown to become asymmetric and thus the mean curvature is not constant. A strong gravitational effect is also manifested by a draining phenomenon where there is continuous reduction in the upper half-ﬁlling angle with increasing separation distance [69]. As a result, there is a difference in the forces acting on the upper and lower spheres, which is equal to the weight of the liquid bridge due to buoyancy. In addition, a bridge ruptures at a smaller separation compared to the gravity free case. When the surfaces of particles are neither smooth and nor chemically homogenous, this leads to wetting hysteresis in which the contact angle is greater than or less than the equilibrium contact angle when the liquid is advanced or retracted, respectively. A characteristic of this phenomenon is that the contact line will remain stationary (i.e. pinning) if the contact angle is intermediate between the advancing and receding contact angles. Once either of these limits is attained, slippage of the contact line will occur. The consequences are that the total liquid bridge force increases when the spheres are separated so that the liquid is retracted with the contact angle being intermediate between the advancing and receding contact angles. Experiments with liquid bridges between spheres have shown that the capillary forces decrease when the receding contact angle is reached allowing the contact line to slip [70]. On reduction of the separation distance, pinning of contact line ﬁrst occurred followed by slippage once the advancing contact angle was reached. This implies an extended bridge rupture distance and that capillary interaction can be dissipative rather than conservative as generally assumed due to the hysteresis in the force–separation curve. Wetting hysteresis is believed to be the contributory factor causing a lower than expected bridge force at near zero separation [70].

2.5.2. Viscous junctions In the above analysis of capillary liquid bridge forces, the particles are assumed to be stationary. The viscous effect of the liquid, however, may be accounted for by the lubrication solution between rigid spherical particles when there is relative normal motion between their centres and provided that the ratio of the separation distance and their radii is sufﬁciently small [71]. Consider the case depicted in Fig. 8, the upper sphere approaches the stationary lower sphere at a normal velocity, nn, causing symmetric radial ﬂow. The parameter nn can be taken as the relative normal approach velocity if both spheres are in motion and would be negative if the spheres were separating. Considering the general case of a power

Mechanistic Description of Granule Deformation and Breakage

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z

vn

B h(r)

Liquid bridge

h0

r

Fig. 8. Schematic diagram showing the squeeze ﬂow between two rigid spheres with the cylindrical coordinate system adopted [72].

law liquid, the shear stress is given by the following relationship: q @v r trz ¼ m0 @z

ð38Þ

where m0 is the ﬂow consistency, q the power law index and vr the radial component of the velocity ﬁeld. The continuity equation based on mass conservation in the integral form may be written as [72] Z h2 2pr v r dz þ pr 2 v n ¼ 0 ð39Þ h1

where the gap may be represented by the parabolic approximation h1 ðrÞ ¼ h01

r2 2R1

and

h2 ðrÞ ¼ h02 þ

r2 2R2

ð40Þ

so that hðrÞ ¼ h1 þ h2 ¼ h0 þ

r2 2R

ð41Þ

Note that h01 and h02 are half-separation distances at r ¼ 0 for the lower and upper spheres, respectively. The pressure gradient in the liquid dp/dr is given by the lubrication approximation as follows: dp @trz ¼ dr @z

ð42Þ

Substitution of equation (38) into (42) with the boundary conditions @vr/@z ¼ 0 at z ¼ 0 (zero shear rate at the mid-plane) and vr ¼ 0 at z ¼ h1(r) ¼ h2(r) (no slip boundary condition) yields the radial velocity distribution. # " q 1 @p 1=q h2 h1 ðqþ1Þ=q hðrÞ ðqþ1Þ=q vr ¼ z ð43Þ q þ 1 m0 @r 2 2

Y.S. Cheong et al.

1080

Combining equations (39) and (43) and integrating between p ¼ pr to patm and r ¼ r to B results in the radial pressure distribution p(r) with respect to the atmospheric pressure patm: Z 2q þ 1 q q B rq pðrÞ ¼ 2m0 vn dr ð44Þ 2qþ1 q r ðhðrÞÞ The total viscous force in the normal direction can be obtained by the surface integral of equation (44) Z 2q þ 1 q q B r qþ2 P vn ¼ 2pm0 vn dr ð45Þ 2qþ1 q 0 ðhðrÞÞ For a Newtonian ﬂuid where q ¼ 1 and m0 ¼ Z is the ﬂuid viscosity, equation (45) reduces to [71] P vn ¼ 6pZðR Þ2

vn h0

ð46Þ

which is consistent with the analysis of Frankel and Acrivos [73]. Numerical solutions and two closed-form approximations for different values of q and B were deduced by Lian et al. [72]. For the force developed in tangential viscous interactions, Pvt, Lian et al. [74] proposed the use of the following asymptotic analytical solution derived by Goldman et al. [75] who studied the motion of a rigid sphere parallel to a rigid wall bounding a semi-inﬁnite viscous ﬂuid 8 R P vt ¼ ln þ 0:9588 6pZRv t ð47Þ 15 h0 vt being the tangential relative velocity of the spheres and R ¼ R for the case of two spheres. They argued that the viscous resistance between two spheres connected by a small liquid bridge is similar to that between a sphere and ﬂat surface at small separations where the ﬂuid resistance in the central inner region between the sphere and the ﬂat dominates.

2.6. Solid bridges It was pointed out that a network of liquid bridges can form within a powder mass or granules as a result of capillary condensation. There are cases where the constituent particles and other contaminants may react chemically with or dissolve in the liquid bridges. Following evaporation of the liquid, the dissolved materials may recrystallise or precipitate to form solid bridges between the constituent particles and thus causing caking. In granulation, the drying of binders in the form of a solution will cause the formation of solid bridges. The solidiﬁcation

Mechanistic Description of Granule Deformation and Breakage

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may result in bridges with complex morphologies and microstructures depending on the type of solution binder employed [76]. The development of the interparticle force between a pair of cylindrical rods of 5 mm diameter connected by a solidifying liquid bridge was monitored by Tardos and Gupta [77]. The measured attractive forces during the solidiﬁcation of pure polymer bridges were two orders of magnitude larger than the initial liquid bridge weight, despite the occurrence of several sharp drops in the measured force indicating the formation of internal voids and cracks during shrinkage of the bridges. Furthermore, it was probable that larger voids were created in larger bridges during solidiﬁcation leading to lower bridge strengths compared with those associated with smaller bridges. In contrast, the crystallisation of liquid bridges containing saturated salt solutions was shown to generate repulsive forces, which were several hundred times the initial bridge weight [77]. Microscopic inspection of the fracture surfaces of granules bound together by these solid bridges revealed that fracture through the solid bridge (cohesive failure) and detachment of the bridge from the particle surfaces (adhesive failure) were both possible. Similar fracture mechanisms emerged from the impact experiments of Subero and Ghadiri [78] using porous glass ballotini granules composed of solidiﬁed polymer binder. The theoretical computation of solid bridge binding forces is not well documented in the literature but the case of cohesive failure was considered by Bika et al. [76]. Although shrinkage of liquid bridges is expected during solidiﬁcation, it was conjectured in their analysis that the bridge proﬁle is preserved with a welldeﬁned neck similar to that shown in Fig. 6. Assuming that bridge fracture occurs across the narrowest cross-section or neck, the normal rupture force Psb is related to the bridge strength ssb [76] P sb ¼ pr 2sb ssb while the bridge neck size rsb is given by " #c r sb CS V b ¼ 2b R rp R3

ð48Þ

ð49Þ

where R is the constituent particle radius, rp the particle density, Vb the initial liquid bridge volume, CS the total concentration of solid dissolved in the liquid, b and c are numerical constants tabulated in ref. [76]. For bridge volumes larger than 0.1 ml, the experiments of Pepin et al. showed that the constants b and c could be approximated as 0.42 and 1/3. To determine the bridge strength, Bika et al. [76] expressed ssb in terms of the macroscopic crushing strength of granules sf using the Rumpf’s theory (see Section 3.1.2) as follows: " #2c 2 1 g C S V b ssb ð50Þ sf ¼ pb g rp R3

1082

Y.S. Cheong et al.

eg being the porosity of the granules. ssb can be evaluated empirically by fracturing macroscopic solid bridges formed between two tablets under three point bend [76]. It was found that the measured values of ssb for the macroscopic bridges agreed reasonably well with those inferred from equation (50) using two model systems of lactose and mannitol granulated with different liquids having different solubilities of the powders. In addition, they pointed out that bridge strength comparable to the tensile strength of the pure polymer binder could be achieved if the constituent particles have low solubility in the binder. However, if foreign substances such as dissolved base powder and surfactant were present, the bridge strength was dependent on the compatibility of the dissolved materials with the liquid binder.

3. MACROSCOPIC GRANULE STRENGTH The strength of a material, on a macroscopic scale, can be interpreted as the resistance to permanent deformation and fracture. To characterise the stiffness, an elastic modulus can be deﬁned as the gradient of the stress–strain response. Typically, for particles and granules, they would be approximated as spheres and force-displacement data from diametric compression would be ﬁtted to the Hertz equation (12). When fracture occurs, it is common to attribute material strength to the maximum stress corresponding to the initiation of crack growth. The value of the stress depends on the intrinsic toughness of the material and the size of the ﬂaw at which fracture starts. For large specimens, it is possible to apply fracture mechanics procedures to accurately measure the toughness (e.g. [79]). However, for single particles and granules it is usually possible to only deﬁne a nominal fracture stress both because of the geometry and because the breakdown mechanisms are likely to be complex, e.g. fragmentation involving multiplecrack pathways. The stiffness and strength of materials are strongly dependent on the evolving surface and bulk stress ﬁelds. For a homogenous elastic sphere (or particle) in contact with an external body, the classical theories of Hertz–Huber [80] and Lurje [81] can be superposed to describe the overall stress distribution within the sphere [82]. Recently, Shipway and Hutchings [83] derived numerical solutions for the elastic stress ﬁelds developed in spheres under uniaxial compression and free impact against a platen. If plastic deformation is initiated, the resulting stress ﬁeld is expected to depart dramatically from the elastic case. It was suggested that Prandtl’s solution [84] might provide a reasonable description of material deformation in a sphere experiencing plastic deformation [85]. Catastrophic failure of solid particles will take place once the maximum allowable stress in the material is exceeded. The failure modes can be classiﬁed into

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three categories, viz brittle, semi-brittle and ductile failures depending on the extent of plastic deformation experienced by the material during fracture. Brittle failure occurs without signiﬁcant plastic deformation whereas substantial plastic deformation can be found in material failing in a ductile manner. An intermediate case where brittle fracture occurs at the boundaries of a small plastically deformed region at the crack tip is termed semi-brittle failure [78]. The above theories provide accurate descriptions of the stress ﬁelds in homogenous non-porous solid particles where local stresses can be transmitted throughout the entire volume of material. Attempts have been made to extend continuum theories of elasticity and plasticity to predict the mechanical response of granular materials to external loads [86,87]. Nevertheless, the predictions have not always been successful as some results are contradicted by experimental observations [88]. Numerous studies have conﬁrmed that heterogeneous stress propagation occurs in particle assemblies such that forces are transmitted through discrete chains of particles [89–91]. The dispute on the limit of continuum mechanics to describe the behaviour of a granular system was resolved recently by the numerical studies of Goldenberg and Goldhirsch [92]. It was demonstrated that the stress response of a granular system resembles that of a continuum solid provided the region of deformation is small compared to the volume of the entire system. In addition, the continuum characteristics are enhanced by increasing levels of friction and disorder of the system. The similarity in the deformation characteristics between granules and continuum solids were noted in the review of Bika et al. [17], despite the inherent disordered structure of granules. From a microscopic perspective, the mechanical properties of an ensemble are dictated by the bonding between the constituent particles. The interparticle bonds in the region of load application may be ruptured causing the particles to shear apart before the load can be transmitted throughout the medium in contrast to a homogenous elastic systems [93]. Thus, it is clear that generally the strength of a granular medium is invariably governed by the interparticle bonding mechanisms rather than the strength of individual constituent particles. Furthermore, the load transmission in a granular body is affected by the internal particle packing. To account for these features, the micromechanical models detailed in the succeeding sub-sections were proposed to estimate the strength of granular materials.

3.1. Micromechanical descriptions of granule strength 3.1.1. Ensemble elastic modulus The elasticity of a particle assembly, which is characterised by the ensemble elastic modulus, is of practical signiﬁcance. For instance, this parameter is the decisive factor in controlling the dimensional change of compacted tablets as a

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consequence of the residual stresses induced during compaction. Moreover, it determines the amount of elastic energy stored in moist granules during collisions encountered in a granulation process. Successful coalescence is promoted when there is insufﬁcient recovery of elastic energy to overcome the bonding developed at the interface between colliding granules [94]. In addition, Kendall and coworkers [32] discussed the possibility of inferring the surface energy of solids from the elasticity of powder compacts. Their work forms a basis for relating the ensemble elastic modulus to the packing and properties of the constituent particles, which is summarised below. The normal contact stiffness, kn, of two autoadhesive spherical particles each with a radius R and identical properties under a small external normal load of P is deﬁned as [32] kn ¼

P PR ¼ 2 d a0

ð51Þ

where a0 is given by "

9pW A R2 ð1 n2 Þ a0 ¼ 4E

#1=3 ð52Þ

which is the JKR contact radius at zero external load. Clearly, the contact stiffness deﬁnition implies the Hertzian force–displacement law instead of the JKR relationship. This inconsistency will be examined later. If autoadhesive particles are stacked vertically together to form a single-particle chain or string with a nominal cross-sectional area of (2R)2 supporting a small tensile load at both ends, the ensemble elastic modulus, E 0 can then be expressed as [32] E0 ¼

nominal stress P=ð2RÞ2 P E ¼ ¼ ¼ a0 strain dð2RÞ 2ð1 n2 ÞR d=2R

Thus, combining equations (51), (52) and (53) leads to " #1=3 9pW A E 2 0 E ¼ 32Rð1 n2 Þ2

ð53Þ

ð54Þ

Since a 3-D ordered cubic array of particles comprises a square packing of these particle chains, it can be shown that the ensemble elastic modulus also follows equation (54), assuming equal contribution of each particle chain to stress transmission. A similar analysis was performed on three other regular packings having denser packing than the simple cubic array, viz, cubic-tetrahedral, tetragonal– sphenoidal and hexagonal arrays. In these arrays, the nominal cross-sectional area is reduced as the particles are packed closer leading to an increase in the coordination number, i.e. additional interparticle contacts are established. Hence, it is necessary to resolve the extra contact stiffnesses in the direction of the load

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application. It follows that there is a rapid rise of the ensemble elastic modulus with the fourth power of the solid fraction of the assemblies such that the following scaling applies [32]: " #1=3 9pW A E 2 0 4 E ¼ 13:3ð1 g Þ ð55Þ 32Rð1 n2 Þ2 In this equation, the air volume fraction of a dry autoadhesive ensemble is deﬁned as the porosity, eg so that the solid fraction f ¼ 1eg. The foregoing analysis was investigated by Kendall et al. [32] using three-point bend tests of beam shaped powder compacts manufactured from submicron zirconia, titania, alumina and silica powders. To produce beam compacts of different solid fraction, each powder was initially mixed with a different amount of aqueous polyvinyl alcohol solutions, which were subsequently removed by heating. It was demonstrated that the measured compact elastic modulus ﬁtted closely to the fourth power dependence on solid fraction. Conformity of the fourth power scaling was also reported in the studies of Trappe et al. [95] and Yanagida et al. [96] using weakly attractive colloidal particles and powders with a broad size distribution, respectively. Hence, it can be concluded that this scaling is applicable to powder compacts with random packing. This correlation was postulated to be the consequence of the f2 dependences of the solid fraction on the density of particle packing and coordination number of each particle [32]. However, the problem associated with equation (55) is that the surface energies of the powders used in the experiments of Kendall et al. [32] were underpredicted with the elastic moduli measured in the bend tests, except for the case of titania. The low surface energy values were explained in terms of surface contamination where surface adhesion was reduced considerably. Nevertheless, Thornton [97] disagreed with attributing surface contamination as the only reason for the low-energy values and pointed out the inconsistency of equation (51) with the JKR force–displacement law. The normal contact stiffness when corrected can be expressed as [52] pﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ E 3 P1 3 PC kn ¼ a0 pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ ð56Þ ð1 n2 Þ 3 P1 PC with P1 and PC deﬁned in equations (17) and (18), respectively. Under zero external load, the normal contact stiffness can be simpliﬁed to k n ¼ 0:6

E a0 ð1 n2 Þ

ð57Þ

which means the ensemble elastic modulus of a simple cubic packing is overpredicted by a factor of 1.67 using equation (55) [97]. Hence, there is an underestimation of the work of adhesion by a factor of 9.26.

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Further experiments of Abdel-Ghani et al. [98] on bend tests of glass ballotini powder compacts highlighted another inadequacy in the model of Kendall et al. [32], which assumed a uniform transmission of the applied stress throughout the entire volume of the ensemble. In their analysis, the work of adhesion for the glass spheres was measured separately and inserted into equation (55). It was found that the ensemble elastic modulus was overestimated by several orders of magnitude compared to the bend test results for the glass compacts. To resolve this issue, Adams et al. [91] proposed that the external load acting on the ensemble is carried via equally spaced discrete particle chains with a spacing factor of s. Combining this approach and Thornton’s [97] analysis, they arrived at the following prediction of the ensemble elastic modulus of a simple cubic packing of autoadhesive particles: " #1=3 2 2:4 9pW E A E0 ¼ 2 ð58Þ s 32Rð1 n2 Þ2 Although s is not known a priori, discrete element simulation data suggested that the typical value of this parameter is of the order of several particle diameters [90]. The prediction of equation (58) was still an order of magnitude larger than the elastic modulus measured by bending glass ballotini beam compacts [98]. This discrepancy could be interpreted more appropriately in terms of foreign molecules contaminating the glass particle surfaces which themselves may also be rough [91].

3.1.2. Rumpf’s theory of granule strength The deﬁnition of strength for particulate media is not universal but determined by the loading conﬁgurations and the failure mechanisms. A granule usually fails in a tensile mode where separation of contacting particles occurs across some failure planes forming macroscopic fragments. Hence, it is appropriate to deﬁne granule strength in terms of the maximum allowable tensile stress acting perpendicularly to the cross-sectional area of a failure plane. In Rumpf’s [99] pioneering theoretical treatment of granule tensile strength, tensile forces were assumed to transmit through a granule body via the bonds between the constituent particles. Different mechanisms of bonding were considered including autoadhesion, liquid bridges and solid binder. To simplify the problem, a granule was assumed to comprise a random distribution of monodisperse spheres where each interparticle bond on the failure plane contributed equally to sustain an external tensile force. The consequent effect is that the failure surfaces separate simultaneously, once the critical tensile stress is exceeded. Summing the interparticle tensile forces across the entire failure plane, the following theoretical relationship for the ultimate tensile strength of granule, sf was

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developed [99] 1 g Fb sf ¼ 1:1 g ð2RÞ2

ð59Þ

eg, R and Fb being the air volume fraction of the granule (granule porosity), radius of the constituent particles and binding force, respectively. Nevertheless, the constituent particles of real granules are often poly-disperse and nonspherical. In respect to this problem, it was proposed that R in the foregoing should correspond to the mean radius, for instance the surface-volume mean radius of the constituent particles [15]. For granules bound together by surface forces, the above expression can be written in terms of the Hamaker constant, A, and the gap between the constituent particle surfaces, z 1 g A sf ¼ 1:1 ð60Þ 48z2 R g This analysis correlates the macroscopic granule strength to microscale parameters, viz the constituent particle properties and their packing density characterised by the granule porosity. There is evidence that equation (59) provides a satisfactory prediction of tensile strength for wet granules when the binding force due to liquid bridge is used in place of Fb as demonstrated experimentally by Rumpf [99], Schubert [100] and Hartley et al. [101] and more recently by the computer simulation of Gro¨ger et al. [102]. However, Pierrat and Caram [103] compared Rumpf’s theory with published experimental data and revealed that the theory overestimates the strength of various types of wet granules in the pendular state, where the liquid content was too low to ﬁll the interstitial spaces. It is possible that these unﬁlled pores possess the stress ampliﬁcation effect similar to a crack. Consequently, these granules might fracture in a brittle manner by rapid crack propagation instead of the simultaneous rupture postulated by Rumpf [99]. Moist granules with pores saturated with liquid are likely to deform plastically resulting in a more uniform stress ﬁeld and hence it is arguable that Rumpf’s theory is applicable. This transition in failure mode was identiﬁed in the experimental investigation of Fu et al. [104] on the impact breakage of wet granules containing different amount of liquid binder.

3.1.3. Kendall’s theory of granule strength Most dry granules and powder compacts fail by brittle fracture where unstable cracks propagate rapidly through the assemblies. Furthermore, granules from identical batches exhibit a distribution of tensile strengths, which are sensitive to the distribution of defects and the critical defect size within the granules. These typical characteristics of brittle fracture were encountered in the studies of

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Kendall et al. [105] on ceramic compacts, Adams et al. [106] on polymer bound sand compacts, Abdel-Ghani et al. [98] on autoadhesive glass assemblies and recently Antonyuk et al. [107] on dry granules. The description of the tensile strength for these particle assemblies using Rumpf’s theory is obviously inadequate as the pronounced effect of defect size was ignored. This is because the stress ﬁeld in a cracked body is highly non-uniform particularly in the immediate vicinity of the defect tip. Hence, the calculation based on the uniform stress distribution assumed by Rumpf [99] is anticipated to overestimate the granule strength since granules are weakened by the high local stresses generated due to the presence of defects and inclusions. A more realistic theoretical tensile strength of a granule can be derived through the summation of the energy required to rupture each interparticle junctions located on the fracture plane [105]. This is in contrast to the force summation employed by Rumpf [99]. Assuming monosized spheres, a fracture energy uf to separate two adhering particles with a contact radius of a0 is expended to create two new solid surfaces. Nonetheless, there is also a simultaneous release of the elastic energy ue due to recovery of the material from deformation under the action of the surface forces at zero external force. Hence, the net fracture energy can be written as [105] uf ¼ pa20 W A ue

ð61Þ

Adopting the JKR adhesion model, the elastic energy with no externally applied force is ue ¼

1=3 2=3 1 4 3ð1 n2 Þ ð3pW A RÞ5=3 15 2R 2E

ð62Þ

whereas a0 is given by "

9pW A R2 ð1 n2 Þ a0 ¼ 4E

#1=3 ð63Þ

so that " uf ¼ 4:736

p5 W 5A R4 E2

#1=3 2 2

ð1 n Þ

ð64Þ

For a simple cubic packing of particles, the number of contacts per unit area across a cleavage plane, i.e. the contact density, (2R)2, whence the total fracture energy Ucubic associated with this plane is given by " #1=3 p5 W 5A 2 2 U cubic ¼ 0:074 2 2 ð1 n Þ ð65Þ E R

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Similar to the case of ensemble elastic modulus, Kendall et al. [105] concluded that the total fracture energy increases rapidly with the fourth power of the solid fraction of regular assemblies. On this basis, the ensemble fracture energy, Uf, can be derived as [105] " #1=3 5 2 2 4 W A ð1 n Þ U f ¼ 59:7ð1 Þ ð66Þ 4E 2 R2 This fourth power scaling was supported experimentally by fracturing beamshaped powder compacts with different solid fractions manufactured from titanium dioxide and aluminium oxide in a three-point bend conﬁguration [105]. However, the values of WA ﬁtted to the experimental data were two orders of magnitude larger than the equilibrium work of adhesion for both powders. This is because energy loss caused by plastic deformation ahead of the crack tip is inevitable when this quantity is measured in fracture experiments [98,108]. Despite the discrete nature of a particle assembly, it can be treated as a continuum body provided the constituent particles are much smaller than the macroscopic size of the assemblage [105]. Applying continuum fracture mechanics, rapid crack propagation in a tensile mode (Mode I) occurs at a critical defect size present in the assembly, c, which can be related to a critical stress intensity factor, KC [109]. pﬃﬃﬃ K C / sf c ð67Þ where Uf ¼

K 2C E0

ð68Þ

Rearranging equations (67) and (68), yields the critical tensile stress for fracture [105] 0 1=2 E Uf sf / ð69Þ c This continuum approach is justiﬁed when the fracture stress obtained by fracturing beam compacts of titania containing through edge notches was observed to decay with c according to equation (69) in three-point bend tests [105]. Notches of different length were deliberately introduced to the edge of the compacts to test the sensitivity of strength to defect size. Although this theory is successful in the case of artiﬁcially cracked compacts, questions were raised about the size of the natural defects. As Kendall [108] demonstrated, the natural defect size present in an unnotched sample was several orders of magnitude larger than the size of the constituent particles. Further analysis revealed that the inclusion of small agglomerated powders in the compact constitutes such defects [16]. This has also been observed by other workers [109].

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Recently, Coury and Aguiar [110] provided an interesting comparison between Rumpf’s and Kendall’s theories by measuring the strength of ﬁltration cake and tumbling drum granules. Their studies revealed that neither Rumpf’s theory nor Kendall’s approach could represent the interparticle interactions within granules found in different applications. Rumpf’s theory was shown to describe adequately the process of ﬁltration cake removal because the cake (granular layer) was formed by slow and uniform deposition of powder. This resulted in layers with low adhesion and uniform porosity, which could be separated simultaneously without crack propagation. However, the granules agglomerated in tumbling drum were observed to be ruptured by propagating cracks under diametrical compression. Hence, Kendall’s approach provided better prediction for the granule strength. This was because tumbling drum granules possessed irregular particle packing and cracks that could be nucleated at voids surrounded by strong interparticle links leading to their subsequent propagation through the granular structures. Therefore, rough estimates of granule strength may be facilitated using the appropriate theories depending on the breakage mechanism in the application concerned. The underlying mechanisms for granule breakage can be revealed by single-granule fracture studies. Some of the experimental investigations and computer simulations of single-granule crushing are summarised in the following sections, with a particular emphasis on spherical dry granules.

3.2. Measurement of granule strength 3.2.1. Diametric compression experiments Quasi-static diametric compression is a common technique for studying the crushing strength of single granules [17,110–112]. It provides a means of measuring the indirect tensile strength of a granule, i.e. the tensile hoop stress required to split the granule apart. This is an appropriate method since real applications usually involve spherical granules rather than beams or bars as studied by Muller et al. [109] and Kendall [108], for example. Furthermore, granule material parameters and the energy utilised to fracture a granule can be estimated from the load–displacement (load–deformation) curve. However, it has been argued that the granule crushing strength measured in diametric compression must be interpreted with great care, as it is only representative for highly brittle and isotropic materials [17]. The deformation characteristics of a dry granule before fracture are strongly dependent on the bonding mechanisms and the defect distribution, which are inﬂuenced by the granulation method. For instance, Sheng et al. [113] showed that polymer bound alumina granules having complex internal structures exhibited a mixture of brittle and ductile behaviour. A variation in crushing strength was reported by Kendall and Weihs [114] and Samimi et al. [115] for granules prepared

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under nominally identical conditions. These observations may be attributed to the susceptibility of granule strength to the structural heterogeneity of granules such as the critical size of the internal voids or inclusions. Antonyuk et al. [107] demonstrated that the deformation behaviour of granules can also be a function of compression speed and granule size. Generally, it is possible to classify the deformation of a granule by inspecting the load–displacement curve obtained by compressing it between two parallel platens. Typically, for dry granules they follow a characteristic elasto-plastic deformation where there exists a non-linear Hertzian regime at small displacements. After a yield limit, where plastic deformation is initiated, the load appears to increase linearly with the displacement of the platens according to the following relationship [116]: P ¼ 2pRHd

ð70Þ

where H is the hardness of the granules which varies from 1.1 sy at ﬁrst yield to 3 sy at full plastic deformation, sy being the uniaxial yield stress of the granules. For the diametric compression of granules where two contact zones exist, the relative approach between the centre of the granule and either platen d is half of the total displacement of the moving platen. Such deformation behaviour was observed in the compression studies of binderless granules produced by spray-drying [114] and ﬂuidised bed granulation [13]. Similarly, granules bound with solid binder such as detergent [115] and zeolite granules [107] were shown to deform elasto-plastically. When the elastic strain is negligible, a granule deforms plastically giving a linear load–displacement relation as exhibited by binderless limestone granules [93] and sodium benzoate granules [107]. Fracture of granules was identiﬁed as corresponding to an abrupt reduction in the platen force. In cases where substantial elastic strain is stored prior to fracture, granules may be disrupted through unstable crack propagation, possibly at sonic speed as suggested by Antonyuk et al. [107]. In contrast, the crack propagation for plastically deformed granules is more stable. Sometimes, partial fracture may occur without total disruption of the granules. This was observed in the compression studies of Sheng et al. [113] on polymer bound granules for which the measured forces ﬂuctuated during deformation. It has been proved useful to treat a granule as a continuum and infer the material parameters from the load–displacement data using the contact mechanics theory outlined in Section 2.3.2.1. The constituent particles of the granule must be small compared to the macroscopic granule size for the continuum approach to be justiﬁed [108]. This was pursued by Kendall and Weihs [114] to determine the elastic modulus and yield strength of spray-dried granules composed of submicron zirconia particles. Since no binding agent was used, the granules were postulated to be bound together by van der Waals attraction. However, the agreement of the inferred elastic modulus with that calculated using a micromechanical model based on autoadhesive particles

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(see Section 3.1.1) was not satisfactory. It was argued that the initial loading of the granules might be a consequence of the irreversible ﬂattening the protruding granule surface rather than the apparent elastic deformation shown in the compression data [114]. More recently, a similar treatment was performed to calculate the material parameters from diametric compression data of polymerbound alumina granules [113], detergent granules [115] and industrial granules [107]. Nonetheless, comparisons with results from micromechanical modelling were not possible due to the complex interactions between solid binder bridges and the constituent particles. Kapur and Fuerstenau [93] investigated the quasi-static behaviour of dry binderless granules by compressing limestone pellets of 8–20 mm in diameter. At fracture, a cone of material was generated at opposing poles of a spherical pellet. These poles corresponded to the contact points between the pellet and the upper and lower plates of the compression device. The general fracture pattern was splitting along a vertical plane creating two hemispherical halves. They suggested that fracture is initiated once the separation between the interparticle bonds along a potential fracture plane exceeded a critical value. Since it was assumed that no preferential fracture plane existed, they proposed that all the interparticle bonds were subjected to the same horizontal tensile force. By equating the total compressive work experienced by the cones at the poles until fracture to the strain energy stored in the thin disk volume encompassing the fracture plane, the crushing strength was related empirically to the pellet porosity and limestone powder surface area. This relationship reﬂected the divergence of the strength of dry, porous pellet from that of homogenous, elastic body. A later quasi-static compression test conducted by Arbiter et al. [117] using large sand–cement spheres (up to 120 mm in diameter) yielded fracture patterns similar to that obtained by Kapur and Fuerstenau [93]. Inspection of the contact area at a load slightly less than that corresponding to the fracture value revealed that minute cracks were densely distributed along the periphery of the contact area. The periphery was indicated by the sharp change in radius of curvature, which was expected to be heavily stressed. The breakage efﬁciency for diametrical compression to produce fragments of a speciﬁc size was deduced and was compared with that for free-fall impact. It was found that the energy input necessary to initiate fracture in free-fall impact was twice that required by quasistatic compression. Moreover, their work indicated that slow compression and low-velocity impact induced geometrically similar stress ﬁelds in spherical sand–cement spheres. The formation of cone was ﬁrst identiﬁed by Newitt and Conway-Jones [118] when wet sand granules fractured under compressive loading. They pointed out that the surfaces of the conical volume of materials coincided with the slip planes in the direction of the principal shear stresses. As a result of the indentation of the cone into the granule body, circumferential tensile hoop stresses were induced,

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which led to the propagation of meridian cracks across the granule. The signiﬁcance of platen friction in cone formation was also emphasised. Referring to the experiments of Arbiter et al. [117], the maximum tensile stresses generated at the perimeter of the contact area might result in the minute cracks observed (see [107]). Apart from strength testing, diametrical compression has been conducted as a complimentary test to visualise the sequence of crack formation in fertiliser granules under impact loading [119]. In this investigation, the load application was continued after fracture to examine the subsequent crack formation. Primary fracture into two hemispheres was observed followed by secondary fracture of the hemispheres into quadrants and segments. Consequently, Salman and co-workers [119] concluded that fertiliser quadrants collected from low-velocity impacts against a platen were the consequence of secondary fracture preceded by primary fracture.

3.2.2. Impact experiments Impact experiments can be used to simulate the dynamic conditions in a granulator and granule handling equipment where granule–granule and granule–wall collisions are involved. Nevertheless, only limited information can be extracted from impact experiments due to the short impact duration. The failure patterns for various types of granules are documented in the comprehensive review of Salman et al. [120], which are summarised in Figs. 9 and 10. In addition to post-impact product examination, several investigators have used high-speed imaging techniques to capture the evolution of an impact event. Arbiter et al. [117] performed a detailed experimental study of the fracture process for large sand–cement spheres (70–120 mm) released from increasing heights above a massive ﬂat steel target. It was observed that cracks dividing the spheres always propagated from the contact region. Furthermore, meridian plane fractures were found to be the dominating failure mechanism at low dropping heights. As the height was increased, there was not only an increase in meridian fracture planes

Fig. 9. Example of the typical impact fracture of the three generic types of granules under moderate impact conditions. These granules are between 4 and 5 mm in diameter [121].

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Fig. 10. Failure forms of solid, wet and binderless granules from the side view. They change with increasing impact velocity from left to right. (After Reynolds et al. [121], with permission.) (a) solid granule; (b) wet granule; and (c) binderless granule.

but oblique fracture planes also started to develop. Eventually, the spheres failed by multiple oblique fractures at the greatest height of fall investigated. Fracture was found to initiate from a conical surface where its base corresponded to the contact area of the sphere at the instant of failure. As this cone was indented into the specimen, the sphere was subjected to tensile hoop stress resulting in meridian crack propagation from the periphery of the contact area. Oblique cracks observed at greater released heights were shown to coincide with the trajectories of maximum compression stress developed when an elastic sphere was in contact with a platen under static loading. Photoelastic experiments were carried out by loading a resin disk with a ﬂat platen under constant loads to simulate the stress distribution within the sand–cement spheres under free-fall impact. From the photoelastic simulation, the stresses were found to concentrate at the corner of the ﬂat portion of the disk as indicated by the maximum fringe order. Thus, Arbiter et al. [117] explained that the failure along the conical surface might start from the periphery of the contact area, as this was the region of high stress concentration. A similar explanation was offered by Salman et al. [119] to interpret the failure patterns observed by impacting smaller fertiliser granules (3.2–7.2 mm) against a rigid massive target. At low impact velocities, meridian plane failure dominated the fracture of the granules into hemispheres or quadrants. However, they concluded that meridian plane cracks were absent at high impact velocities as indicated by the largest surviving fragment, which showed rotational symmetry. In addition, they also found a crushed cone around the contact region in contrast to

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the undamaged cones generated in the free-fall impact tests of Arbiter and coworkers [117]. At the highest impact velocity employed, disintegration occurred leaving a cone of compacted powder on the target surface. A 2-parameter Weibull equation was applied to characterise the probability of breakage for granules impacted at a speciﬁc impact angle and impact velocity. For the impact of these fertiliser granules at various angles, Maxim et al. [122] deﬁned the critical impact failure velocity as the velocity to cause 63.5% of the granules in a batch of nominally identical samples to fracture. Adopting a quasi-static approach, they assumed that the maximum force experienced by the granules before fracture was given by the Hertz equation (12). Integration of equation (12) in dimensionless form allowed the normal component of the critical impact failure velocity, Vf to be expressed in terms of the fracture force and the material properties of the granules [122]. A similar relationship was derived by Knight et al. [123] for the impact of steel spheres against glass surfaces, in which the kinetic energy of an impacting steel sphere was assumed to be converted into the strain energy stored in the sphere at the instant of maximum compression. Maxim et al. [122] demonstrated that the fracture force could be estimated by compressing the granules diametrically up to fracture and the values of Vf calculated on this basis were in reasonable agreement with those measured in impact experiments. In the high-speed imaging investigation of Subero and Ghadiri [78], different breakage behaviour was exhibited when 30 mm granules with a polymer binder containing artiﬁcial macro-voids were impacted against a target plate. It was interesting to note that a cone was not generated at the impact site in any of their experiments. Local disintegration prevailed at low impact velocities and, consequently, particles adjacent to the impact site detached from the main structure of the granule forming singlets, doublets and triplets. Moreover, there was an increase in the extent of disintegration at higher impact velocities. This was considered to be the effect of macro-voids as the presence of these voids at the impact location hindered the transmission of stresses through the granular structure. However, fracture combined with local disintegration might occur when the impact velocity was raised depending on the internal structure of the granule. In this case, signiﬁcant stresses could be transmitted to the bulk of the granules. Thus, cracks were available to split them into several large clusters by side chipping or a combination of meridian and oblique crack fractures. It was found that if fracture took place, the accompanying local disintegration was less severe compared to the cases for which fragmentation was not observed. Generally, it was concluded that the fragmentation of a granule could be promoted by increasing the impact velocity or macro-void size and number. In the impact crushing of 150 mm concrete spheres, fractures along meridian planes with the fragmentation of the conical region at the impact site into ﬁnes were reported by Tomas et al. [124] in contrast to the intact cone observed by Arbiter et al. [117]. This was explained as the result of plastic deformation at the

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impact site causing local crushing. Furthermore, meridian cracks were initiated along the periphery of the contact area of the spheres due to the development of tensile stresses, which eventually propagated and divided the spheres into segments. However, it was suggested that a circular crack circumscribing the contact area would have formed if the spheres experienced elastic deformation at the impact site. The convergence of the circular crack as it propagated into the spheres might lead to the formation of a coherent cone. Besides these macroscopic failure modes, the failure mechanism of interparticle bonds was examined for glass ballotini granules bound with a polymer using scanning electron microscopy in the work of Subero and Ghadiri [78]. Two different interparticle bond failure mechanisms were identiﬁed namely adhesive and cohesive breakages. Adhesive breakage refers to the separation of contacts at the interface between the binder and primary particle resulting in a smooth fracture surface. Internal failure across the binder bridge with rough and irregular fracture surfaces is termed cohesive breakage. However, a correlation between the bond fracture toughness and the macroscopic granule impact breakage behaviour was not reported. A completely different failure pattern was described by Boereﬁjn and co-workers [12] when studying the impact breakage of weak lactose granules. Their results indicated that the disintegration of these granules was due to the bulk plastic deformation induced in the agglomerates. However, embrittlement of the lactose granules was found in a humid environment. In the presence of moisture, the transformation of amorphous lactose to the monohydrate state was initiated, which was postulated to cause the formation of brittle solid bridges. Extensive experimental investigations were carried out by Samimi et al. [125] to examine the inﬂuence of the structural characteristics of detergent granules on the impact breakage propensity. Since the granules contained a viscous binder, humidiﬁcation and storage temperature were found to modify the bonding mechanism between the constituent particles, thus, giving different extents of breakage. In addition, the manufacturing route was demonstrated to be important in determining the granule strength by tailoring the degree of consolidation of the granules.

3.2.3. Multi-granule testing It is clear that mechanical testing of single granules is informative in revealing the complete stress–strain behaviour and the associated failure mechanisms. However, the disadvantage of such tests is that a large number of experiments are necessary to obtain statistically reliable granule strength. A robotic tester was developed by Pitchumani et al. [126] to automate single-granule compression tests. However, from the standpoint of monitoring the degradation of granular

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product in handling equipment, it may be sufﬁcient to express the strength of a bulk granule sample in terms of an average value. In this case, strength measurements can be performed by stressing multiple granules simultaneously. To avoid tedious single-granule crushing tests, Adams et al. [127] proposed a granule strength measurement using a uniaxial conﬁned compression of a bed of granules. Based on the Mohr–Coulomb failure criterion, the shear strength of a single granule was related to a constant of proportionality determined from the linear portion of the pressure–volume compaction curve of the granule bed at large strain. The inferred failure force was of the same order of magnitude as the crushing load measured separately by fracturing single-polymer-bound silica granules under diametric compression. Besides the application of a slow compression velocity, multiple granules are commonly subjected to vibratory motions to assess the breakage propensity. For example, Beekman et al. [128] investigated repeated impacts (fatigue failure) of a group of enzyme granules enclosed in containers attached to a sieve shaker. In their experiments, the enzymes granules were weakened by the fatigue loading resulting in either attrition or gross fragmentation. In the study of Utsumi et al. [129], the strength of fragile ferric oxide granules was examined by agitating the granules on a sieve of a known size using a shaker capable of imposing rotating and vibrating motions. The extent of breakage was expressed as the sum of abrasion rate of the granule bed on the sieve surface and the wear rate of the granules within the bed.

3.2.4. Quanti¢cation of breakage propensity In single-granule fracture experiments, the analysis of failure patterns and the fracture surfaces of the resulting fragments provides insights into the failure mechanisms. In addition, the extent of breakage can be assessed by examining the size distribution of the fragments and fragment size distributions serve as the breakage function for population-balance modelling of a rate process. Nevertheless, there are not standard guidelines available to determine the size of a fragment. This is because most fractured products are irregular in shape and the fragment size depends on the orientation of the fragment under inspection [130,131]. For example, the probability of an elongated fragment passing through a certain sieve size is determined by the orientation of the longest axis of the fragment. Therefore, it is a common practice to represent the size of a fragment as a linear dimension at the particular orientation during inspection. This is usually expressed in terms of equivalent diameter of a sphere, such as equivalent volume diameter [130]. After deﬁning the size of fragments, the distribution of fragment sizes over a range of sizes can be expressed in several forms of empirical relationships. In the

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comminution and fracture of brittle solids, the commonly employed relationship is the Rosin–Rammler equation. It is deﬁned by two empirical constants n and xc namely the width of the fragment size distribution and mode size, respectively [132]. Its cumulative form can be written as follows: n x Q ¼ 1 exp ð71Þ xc where Q is the cumulative probability of a fragment having a size smaller than x. This distribution function is useful in describing skewed-size distributions of comminuted products as pointed out by Djamarani and Clark [133]. The theoretical analysis of Gilvarry and Bergstrom [134] suggested that the parameter xc can also be interpreted as the spacing between the defects within a specimen. This proposal was based on the fact that the fracture of brittle solids is a consequence of crack initiation at a defect site. In addition, Cheong et al. [135] pointed out that n could be used to indicate the fracture form based on their investigation of the fracture of glass spheres impacting a target at different velocities and impact angles. The cumulative fragment size distribution can also be plotted on a Gates–Gaudin–Schuhmann double-logarithmic plot [117,131]. Two distinct straight lines with different slopes distinguishing the debris from coarse fragments are usually observed. The complete size distribution of fragments is necessary to describe the extent of breakage if fragmentation occurs. In the case of the attrition and the wear of granules, ﬁne debris is generated leaving a large intact residue. The analysis of the size of individual debris is difﬁcult since ﬁne particles adhere strongly to each other. According to Boereﬁjn et al. [12], the breakage of weak lactose granule under impact can be quantiﬁed as the proportion of the mass lost by the feed granules after impact. The large residues were separated from the debris by a sieve of a certain size smaller that the feed granules. Attributing all handling loss to the residues, Boereﬁjn et al. [12] deﬁned the lower limit of the breakage extent as x ¼

Md 100% Mf

ð72Þ

while the upper limit, where all handling loss was attributed to the debris, was taken as xþ ¼

Mf Mr 100% Mf

ð73Þ

with Md, Mr and Mf being the debris mass, residue mass and feed granule mass, respectively. This breakage parameter was shown to be directly proportional to the impact kinetic energy since the data scale with the square of the impact velocity.

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3.3. Discrete element method (DEM) simulations Recently, computer simulation has been a robust tool to study the deformation and breakage of particulate systems. This is because the same particle conﬁguration can be tested repeatedly and information about different parameters at any instant of time can be retrieved for further analysis. Furthermore, computer simulation offers the advantage of revealing information, such as different energy dissipations or load transmission paths, which are not accessible through physical experiments. The DEM is a suitable tool for studying the macroscopic response of a particulate system, which depends on the discrete behaviour of its constituent primary particles. This is in contrast to ﬁnite element simulations where the average local stress response of the material is described using constitutive relationships. The study of granule breakage mechanisms under external forces is one of the examples of the application of DEM simulation. The evolution of granule damage is modelled as a dynamic process by tracing the motion of the constituent particles throughout the impact event using Newton’s law of motion. The resulting particle motion is determined by the interaction at the interparticle contacts in addition to external forces such as gravity. The simulation is advanced over a large number of small-time steps and the particle motion is updated continually. This methodology was initially proposed by Cundall and Strack [136]. Attempts to study granule degradation were initiated at Aston University in the UK by incorporating well-established contact mechanics interaction laws into the methodology of Cundall and Strack [136]. The simulations were ﬁrst carried out in 2D [137] and later extended to 3D by Ciomocos [138]. They are now capable of simulating the interactions between elastic, spherical, frictional and autoadhesive particles. The earlier version of the code by Thornton and Yin [52] considered only elastic deformation at autoadhesive interparticle contacts. Plastic yield was accounted for in the subsequent version developed by Thornton and Ning [139]. A slightly different approach was adopted by Potapov and Campbell [140] to represent an elastic solid by ‘‘gluing’’ polyhedral elements together. The glue at the interface between two elements could withstand certain tensile stresses before fracture occurred. Using this modiﬁed technique, correlation between the breakage patterns of an elastic solid and different fracture mechanisms was established.

3.3.1. Diametric compression simulations There are only limited DEM simulations for the diametric compression of granules, presumably because substantial computing time is required owing to the low compression speeds employed. In the study of Thornton et al. [141], distinctive breakage behaviour was observed depending on the porosity and thus the

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packing density of the granule being investigated. They concluded that the fracture of a granule across a well-deﬁned surface is only possible if the granule structure is sufﬁciently dense and rigid, i.e. high density of interparticle contacts. This allows the compressive loads to be transmitted along a discrete path into the granule body from the compression platens. Consequently, sliding between the particle clusters adjacent to this force transmission path occurs leading to the ultimate fracture. This may explain the phenomenon in which dense granules always fracture whereas loosely packed granules fail by progressive displacement of the constituent particles from the contact zones. In addition, two highly fragmented zones were formed at the contact zones when a dense granule fractured along a primary meridian plane in the diametric compression test simulated by Thornton et al. [141]. Similar breakage behaviour was also noted by Khanal et al. [142] in their DEM simulations of diametric crushing of concrete spheres. This is consistent with the cone formation and diametric fracture reported in the slow compressions of wet granules [118] and dry limestone pellets [93], described previously. Finite element computation suggested that the diametric crack should propagate from the peripheries of both contact zones of the granule, where a high tensile stress existed [142]. Recent advancement in X-ray microtomography has enabled Golchert et al. [143] to incorporate real granule structures into DEM simulation to study the effects of granule structure and shape under compressive loading. Their results demonstrated that a spherical granule was more resistant to damage compared to a granule of irregular shape. Again, the breakage behaviour was determined by the probability of forming a strong force transmission path.

3.3.2. Impact simulations Besides diametric compression simulations, the Aston code has been used to investigate granule impact breakage against a plane surface [131,144–147]. It appears that the resulting breakage patterns are very similar to those generated in diametric compression tests, except that there is only one stress application point in impact. The recent review of Mishra and Thornton [148] reported that there are ﬁve factors governing the breakage behaviour of granules under impact. They are the impact velocity, bond strength (interface energy), granule porosity, coordination number of the constituent particles and the local structural arrangement of particles near the impact region. Investigating the combined effects of impact velocity and porosity, signiﬁcant breakage was found to occur when the impact velocity exceeded a certain threshold value. Once breakage took place, dense granules always fractured while loose ones disintegrated. Those with intermediate porosities exhibited mixed-mode failure where both fracture and disintegration occurred. Furthermore, they compared the breakage behaviours

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between similar granules, one with a greater particle contact density than the other. The denser granule fractured in contrast to disintegration shown by the granule with the lower contact density. It was postulated that signiﬁcant stresses were transmitted through the bulk of the granule with the higher contact density storing sufﬁcient elastic energy for fracture. They also suggested that different breakage patterns could be obtained when different parts of a given granule were subjected to impact. This was due to the differences in the local particle arrangement near the impact location. Similar observations emerged from the DEM simulations of Moreno et al. [149] where a granule was impacted at various oblique angles on to the target plane. Moreover, the component of the impact velocity normal to the target plane was found to be the main cause of granule damage under oblique impact. Using the same code, Kafui and Thornton [150] simulated the collision between a pair of similar granules to understand the fragmentation process due to the impact arrangement. In DEM simulations, the extent of breakage can be characterised by the size distribution of the fractured products in a similar manner to that performed in physical experiments. Depending of the intensity of the applied stresses, the fragments can be several clusters of constituent particles that remain intact after a breakage event and some single constituent particles. The size of a cluster can be deﬁned using an equivalent diameter of a solid sphere having the same volume as the cluster [131,147]. Subero et al. [147] attempted an experimental veriﬁcation of their DEM simulations of granule–wall collisions by constructing physical granules using glass beads bound together with a polymer binder. Using the deﬁnition of fragment size deﬁned above, it was found that the fragment size distribution resulting from the simulation was comparable to the experimental results in the complement region, i.e. the fragment size range smaller than 50% of the original granule diameter. However, it should be noted that disintegration of granules was observed in the simulation of Subero et al. [147] rather than the fracture behaviour shown by their physical granules. In a study of the impact breakage of weak lactose granules, Boereﬁjn et al. [12] calculated the lower and upper limits of the granule breakage in physical experiments using equations (72) and (73), respectively. Their parallel DEM results indicated that the actual extent of breakage was within these limits conﬁrming the robustness of the DEM methodology. In another numerical study, Kafui and Thornton [131] reported that the exponent of the power law relationship describing the fragment size in the complement region was a function of the interparticle bond strength. The damage ratio was ﬁrst proposed by Thornton et al. [144] as an alternative measure of the extent of breakage of granules in DEM simulations. This ratio is deﬁned as the proportion of interparticle bonds broken within a granule due to a breakage or deformation event compared with the initial state of the granule. Further work showed that the damage ratio is proportional to a dimensionless group known as the Weber number, Wb [144,147], which takes the following

Y.S. Cheong et al.

1102

form: Wb ¼

2rp V 2i R WA

ð74Þ

where r, R, WA and Vi are the density and radius of the constituent particles, the work of adhesion required to separate an interparticle contact and the impact velocity of the granules, respectively. However, it is uncertain if these parameters should be ascribed to the properties of the bulk granules or those of the individual constituent particles [131]. Furthermore, an asymptotic limit of efﬁciency was identiﬁed beyond which no further breakage is possible with additional energy input [142,147].

4. PRACTICAL CONSIDERATIONS 4.1. Understanding wet granulation mechanisms In a practical context, it is desirable to assess the inﬂuence on the kinetics and growth mechanisms of a wet granulation process due to the modiﬁcation of an existing formulation or the use of a new combination of particle and binder systems. Although satisfactory quantitative predictions are not available currently, it has long been established that granule deformability has a strong effect on granule growth behaviour [151–153]. In most wet granulation processes especially in a mixer granulator, granules collide between each other and the equipment walls. These impact events may result in either rebound or coalescence and sometimes fracture of the granules, which controls the balance between granule growth and size reduction. In a collision, granules that deform easily will form a large area of contact as the impact kinetic energy is dissipated forming a stronger bond that is more likely to outlive ensuing collisions and thus resulting in successful granule coalescence. Conversely, strong granules, which are less deformable, form a weaker bond that will be easily broken apart by subsequent collisions and thus limiting granule growth. The following discussion of granule strength is restricted to that of wet granules where the interstitial spaces between the primary particles are occupied by liquid and air. The collision between a pair of granules is difﬁcult to investigate directly, so a more controlled granule–wall impact conﬁguration is often used as a step towards understanding granule impact and rebound behaviour. The deformability of a granule impacting a plane surface can be expressed in terms of the contact ratio, i.e. the ratio of the deformed area between the granule and the surface to the original cross-sectional area of the undeformed granule. This contact ratio can be related to the dynamic strength of the granule such as the dynamic yield stress [153]. Given the rate-dependent viscous dissipation of the liquid binder within the

Mechanistic Description of Granule Deformation and Breakage

1103

granule during impact, the dynamic strength is sensitive to strain rate when a critical value is exceeded [154–156]. With respect to granule coalescence, the coefﬁcient of restitution can be a useful parameter for predicting the aggregation efﬁciency that is the proportion of granule and/or primary particle collisions that result in an aggregation event. The coefﬁcient of restitution is the ratio of the rebound and impact velocities. For example, a value of zero would imply that the colliding granules would stick together rather than rebound. The breakage of granules can also be expressed in terms of the critical impact velocity to cause crack initiation during impact [104]. The amount of deformation that is experienced by granules in a granulator can be expressed in terms of the Stokes deformation number, Stdef as follows [3]: St def ¼

rV 2rel 2Y

ð75Þ

where Y is the dynamic yield stress, r the granule density and Vrel the relative velocity between the impacting granules. Iveson et al. [154] demonstrated that the mechanical strength of granules is strain rate dependent when a critical compression speed was exceeded. Arguably, it is more realistic to measure the dynamic yield strength under the impact conditions expected in the granulator of interest. The Stokes deformation number is also used to demarcate different regimes of granule growth (see Fig. 11). Granules that do not deform easily during a collision have a small value of Stdef and will only begin to coalesce and grow if liquid binder is expelled to their surfaces due to consolidation to bind the granules together; such systems are termed induction growth systems [3].

"Dry" FreeFlowing Powder

0.1

"Crumb"

Slurry/ Over-Wet mass Steady Growth Increasing Growth Rate

Increasing Deformation Number, Stdef = gUc2/2Yg

Nucleation Only

f(Stv)

Induction Decreasing Induction Time

0

100% Maximum Pore Saturation, smax = ws(1-min)/| min

Fig. 11. Proposed modiﬁed regime map. The nucleation-to-steady growth boundary and steady-growth-to-induction-growth boundaries are functions of Stdef and there is no distinct rapid growth regime [3].

Y.S. Cheong et al.

1104

Binder Viscosity (mPa.s)

10,000 Storkes Number Analysis 1,000 Granules Formed 100

Paste Formed

10

1 0

50 100 150 200 250 Median Size (microns) of the Constituent Particles

Fig. 12. Binder viscosity vs. median particle size showing regions in which granules did and did not form for the agglomeration of glass ballotini with silicone oils in a high shear mixer. Line shows prediction of equation (76) [157].

However, granules with intermediate values of Stdef deform easily during a collision to form a larger contact area leading to steady growth. However, granule breakage is induced at greater values of Stdef. Keningley et al. [157] also developed a strain criterion which describes whether a wet granule will break or survive in high shear granulation, depending on the amount of strain resulted from the compression during the impact. Assuming that granule deformation depends on the pressure loss through the ﬂowing viscous ﬂuid between particles upon impact, the collisional kinetic energy can be equated to the plastic deformation energy of the granule to obtain the following: 2m ¼

3g rD uo d 3;2 1 540 1 2g m

ð76Þ

where em is the maximum compressive strain, eg the granule porosity, rD the granule density, uo the granule impact velocity, d3,2 the sauter mean diameter, m the binder viscosity. They mentioned that the granules will break when the maximum strain, em, exceeds 0.1. This model allows a plausible interpretation of the effect of binder viscosity and primary particle size on the ability to form granules during high shear granulation (Fig. 12) and the predicted limit is shown to be reasonably consistent with the experimental data. Experimental studies have indicated that the impact deformation and breakage of wet granules are governed by various process and formulation-related factors, as described in the following subsection.

Mechanistic Description of Granule Deformation and Breakage

1105

Restitution coefficient

0.25 S=0.115

0.2

S=0.135 S=0.165

0.15 0.1 0.05 0 0

5

10 15 Impact velocity, m/s

20

Fig. 13. The variation of the coefﬁcient of restitution with impact velocity for calcite granules with different binder contents. (After Fu et al. [156], with permission.)

4.1.1. E¡ect of impact velocity In a granulation process, the velocity of granules is dictated by the type of granulator and the operating conditions. The variation of the coefﬁcient of restitution (granule deformation) with impact velocity was investigated by Fu et al. [104] for spherical calcite granules with various amount of liquid polyethylene glycol (PEG 400) colliding with a plane surface. According to the typical proﬁles depicted in Fig. 13, the coefﬁcient of restitution curves attained maxima at some intermediate impact velocities with sticking at smaller and greater values of two critical velocities. The lower limit arises because the surfaces of the granules were covered by a layer of liquid binder. The release of the elastic strain energy stored during impact is insufﬁcient to rupture the liquid junctions formed between the granule and the target. It was argued that the rising parts of the curves were a consequence of the elastic recovery of the granules being sufﬁcient to overcome the viscous dissipation due to the liquid junctions, since the stored elastic energy increases with velocity in the elastic regime [104]. At high impact velocities, the coefﬁcient of restitution was observed to decay approximately with an inverse power of impact velocity as a result of plastic deformation. It is worth noting that such decay was more rapid than that expected from contact mechanics calculations assuming that the granules are treated as homogenous elasto-plastic materials [104]. No improvement in the predictions was found by accounting for ﬁnite plastic deformation and strain hardening effects in the contact mechanics formulations [158]. Arguably, the contact deformation of the

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Y.S. Cheong et al.

Fig. 14. Dynamic yield stress as a function surface mean particle size for glass ballotini with water and glycerol binders. (Adapted from Iveson and Litster [153], with permission.)

granules at high impact velocities exceeded the limits allowable in quasi-static contact mechanics. Despite the discrepancy associated with wet granules, the coefﬁcient of restitution of granules held together by solid bonds was consistent with the prediction of elasto-plastic contact mechanics with strain hardening [158].

4.1.2. E¡ect of primary particle size Fu et al. [156] showed that the coefﬁcient of restitution decreases with increasing primary particle size for the granule types that were investigated. Although it was acknowledged that the distribution of primary particle sizes can complicate experimental results, this effect was concluded to be due to an increase in interparticle contacts, and hence the density of interparticle forces. Similarly Iveson and Litster [153] found a decrease in the dynamic yield stress of agglomerates with an increase in primary particle size, when water was used as a binder (see Fig. 14). They proposed that decreasing particle size decreases the average pore size between particles and increases the volume density of interparticle contacts. This increases both the capillary and the interparticle frictional forces and thus explains why the yield stress increases when water is used as the binder compared with glycerol. However, they also considered why there was not a signiﬁcant inﬂuence of particle size on the dynamic yield stress when a more viscous binder (glycerol) was used as the binder. Their arguments were based on lubrication theory, which predicts that the viscous force to dominate as interparticle spacing decreases, thus increasing the yield stress.

Mechanistic Description of Granule Deformation and Breakage

1107

Fig. 15. Relationship between tensile strength, st, and wet granule saturation, S. (After Schubert [100], with permission.)

4.1.3. E¡ect of binder content, binder viscosity and binder surface tension Schubert [100] investigated the relationship between tensile strength and the saturation of wet granules. As shown in Fig. 15, he argued that it would depend on the granule state as described in [159]. In the ﬁgure, Sp corresponds to the liquid saturation at which there is a transition between the pendular and funicular states, and Sc denotes the lower limit of the capillary state. In the pendular state, a binder forms discrete lens-like rings (liquid bridges) at the points of contacts between particles, leaving air as a continuous medium. The capillary state describes the completely saturated granule and the funicular state is intermediate between these two states. The tensile strength is expected to increase monotonically with increasing saturation in the funicular state (SpoSoSc) as both bridge bonding and bonding caused by regions ﬁlled with liquid contribute to the tensile strength. In addition, the tensile strength is then expected to decrease at high levels of saturation, as the material becomes a paste. Schubert [100] found reasonably close agreement between experimental observations and this hypothesis. However, this is a problem of considerable complexity given that there are not 3-D analytical solutions available for the forces developed in the funicular state and that it is necessary to account for the inﬂuence of the capillary forces on the friction at interparticle contacts, in addition to the capillary and viscous forces generated by deforming the liquid junctions as will be discussed below. Iveson and Litster [153] examined the dynamic yield stress of cylindrical pellets made from either 19 or 31 mm ballotini with water, glycerol and surfactant binders. They found that with glycerol as the binder, increasing the binder viscosity from 0.001 (for water) to 1 Pa s greatly decreased the amount of pellet deformation as shown in Fig. 16. They attributed this observation to an increase in viscous dissipation. However, they also observed that the effect of binder content is complex. With water as a binder, at low moisture contents, increasing the amount increased the yield stress. However, at greater moisture contents, this resulted in a reduction in the yield stress, whereas with glycerol they found that increasing

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Fig. 16. The dynamic yield stress as a function of binder content for pellets made from two different sized glass ballotini with water, glycerol (Gly.) or NDBS surfactant solutions. (Adapted from Iveson and Litster [153], with permission.)

the amount always caused the yield stress to increase monotonically. They argued that this complex behaviour is a result of a balance between the interparticle friction, capillary and viscous forces that all resist granule deformation. For example, they suggested that increasing the binder content can reduce interparticle friction by lubrication, whereas capillary forces are increased up to the saturation point. In addition, the viscous forces were considered to increase with increasing binder as more would be squeezed between the primary particles. They suggested that for low viscosity binders, an increase in binder content will increase the capillary forces and hence the yield stress. However lubrication effects should eventually dominate, resulting in a decrease in strength at higher binder contents. For binders with a higher viscosity, the viscous forces should dominate, and hence an increase in binder content will increase the agglomerate yield stress (up to the point at which the agglomerate becomes a slurry). Recently, Fu et al. [156] found that increasing the binder content of wet calcite granules resulted in an increase in the kinetic energy dissipated on impact and hence leading to a reduction in the coefﬁcient of restitution (see Fig. 13). By diluting glycerol with water, calcite granules containing the same amount of liquid binder but with different viscosities were prepared. As shown in Fig. 17, the coefﬁcient of restitution of these granules was found to decrease monotonically with increasing binder viscosity. This was attributed to the effective lubrication by the more viscous binders and so reducing the interparticle friction. However, it should be noted that the magnitude of the friction does not directly relate to the energy dissipation. Actually, high interparticle friction stabilises an assembly resulting in less energy dissipation [160].

Mechanistic Description of Granule Deformation and Breakage

1109

0.25

Restitution coefficient

0.2

0.15

0.1

0.05

0 0

500 1000 Binder viscocity, mPa s

1500

Fig. 17. Relationship between binder viscosity on the restitution coefﬁcient at an impact velocity of 5.86 m s1 made from Durcal 15 (calcite) and a binder ratio of 0.15 [156].

4.1.4. Remarks Despite the promising predictions of the growth regime maps depicted in Figs. 11 and 12, the effects of binder properties on the impact deformation of wet granules are strongly dependent on the systems being investigated. Iveson and Litster [161] commented that even a qualitative prediction of the impact deformation of wet granules and the resulting growth behaviour required a knowledge of the relative contributions of the interparticle friction, capillary and viscous forces. To resolve this problem, it is possible to employ the discrete particle computer simulations of the type described by Lian et al. [74] where the interparticle interactions are parameterised by the particle properties and interfacial interaction between the particles and liquid binder. However, this work was limited to the pendular state for which closed-form interactions laws are available.

4.2. Targeted performance For many granulated products, it is necessary to re-disperse the constituent particles in liquid media for applications involving, for instance, coffee or detergent granules. When two contacting particles without a binding agent being present are immersed in liquid, an additional force known as the solvation force co-exists with the van der Waals and possibly electrostatic forces acting between the particle surfaces. It has been found that this solvation force is repulsive for hydrophilic surfaces, whereas attractive solvation forces operate between hydrophobic surfaces in the case of water [21]. Readers are referred to the comprehensive review of Israelachvili [162] for further information on the

Y.S. Cheong et al. 80

100

70

90 Contact angle (°)

Surface tension (mN/m)

1110

60 50 40 30 20 Pure IPA

10

70 60 50 40 30

0 0

(a)

80

2

4 6 8 10 12 IPA concentration (% v/v)

14

16

35

(b)

40

45 50 55 60 65 Surface tension (mN/m)

70

75

Fig. 18. The variations of (a) the surface tension of the aqueous IPA solutions and (b) the contact angle of the solutions on a polystyrene substrate with increasing concentration of IPA [163].

molecular forces acting on particles in liquids. It follows that the dissociation of granules occurs in a liquid if the solvation forces acting on the constituent particles are repulsive. Hence, the dissociation process can be correlated to the interfacial interactions between the liquid medium and the particle surface. In the experimental work of Adams et al. [163], a reduction in granule fracture strength was observed when binderless granules composed of autoadhesive polystyrene particles were compressed diametrically in liquids of different surface tension (see Fig. 19(a) for results). The liquids were prepared by mixing different amounts of isopropanol (IPA) with pure water to vary the surface tension from ca. 71 to 40 mN m1 as shown in Fig. 18(a). The interfacial interaction between the liquids and the polystyrene particles was determined by measuring the contact angles of the liquids on a compact of the polystyrene particles. It may be seen from Fig. 18(b) that the contact angle decreased progressively with increasing concentration of isopropanol in the aqueous alcohol solutions. The contact angle for pure water on the polystyrene surface was found to be approximately 931 as would be expected for a hydrophobic surface. A reduction in the contact angle implied an increase in the spreading tendency of the aqueous alcohol solutions at higher concentration of isopropanol due to stronger solid–liquid interaction. According to the proposal of Ottewill and Vincent [164], it is believed that the isopropanol molecules adsorb onto polystyrene surfaces with the alkyl chains (CH3CH2CH2–) directed towards the particle surfaces at low isopropanol concentrations. This is because dispersion interactions are favoured between the polystyrene surfaces and alkyl chains, which are both hydrophobic. Under such circumstances, the hydroxyl groups of the alcohol molecules are exposed to the solution phase offering a hydrophilic surface to establish hydrogen bonding with water molecules. This implies that the aqueous isopropanol solutions were effectively interacting on a surface with increasing hydrophilicity capable of polar interactions, as the concentration of isopropanol was increased. Hence, spreading of the solutions was promoted as manifested by a reduction in the contact angle.

Mechanistic Description of Granule Deformation and Breakage

1111 Isopropanol molecule

Fracture stress (kPa)

40 Curved surface of a particle 30 A 20

B

10 Curved surface of a particle

0 40

(a)

REPULSIVE FORCES

45

50

55

60

65

70

75

Water molecule

(b)

Surface tension (mN/m)

Fig. 19. (a) The variation of the fracture strength of polystyrene granules compressed diametrically in aqueous IPA solutions of reducing surface tension (adapted from Adam et al. [163]) and (b) a schematic diagram illustrating the origin of the repulsive solvation forces (adapted from Israelachvili and McGuiggan [165]).

It may be noticed from Fig. 19(a) that there is a weak inﬂuence on the fracture strength of the liquids with surface tensions greater than approximately 42 mN m1. A profound effect is evident when the liquid surface tensions were reduced to less than 42 mN m1. The granules disintegrated almost instantaneously into clusters of constituent particles without the input of mechanical energy when immersed in 15% v/v aqueous isopropanol solution with a surface tension of 39 mN m1. In a ﬂuid environment, the autoadhesion between the constituent particles of the granules is characterised by the solid–liquid interface energy gSL instead of the solid surface energy of polystyrene gS. Hence, the work of adhesion for two adhering solid surfaces in a liquid medium WA0 is given by [19,166,167] W 0A ¼ 2gSL

ð77Þ

and rearrangement of equation (4) gives gSL ¼ gS gLV cos yE pe

ð78Þ

The above study [163] was complicated by the fact that adsorption of the isopropanol molecules on the polystyrene particle surfaces is possible, as discussed with respect to the contact angle measurements. Consequently, it is not justiﬁable to ignore pe to obtain a quantitative variation of the interparticle adhesion, i.e. gSL with the surface tension of the aqueous isopropanol solutions based on equation (78). Qualitatively, the fracture strength remains relatively constant when the granules were immersed in liquids with surface tensions greater than about 42 mN m1. Hence, this may imply that there is insigniﬁcant variation of gSL over this range of liquid surface tensions. Two effects of the presence of ﬂuid molecules on the attenuation of the autoadhesion between solid surfaces were discussed by Kendall [20]. The ﬁrst is relevant to the adsorption of ﬂuid molecules on the solid surfaces due to

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spreading. The consequent result is that a thin ﬁlm of ‘‘contaminant’’ is deposited, which prevents direct molecular contact between the solid causing a reduction in adhesion between the surfaces. This is consistent with the interpretation given above that immersion of the polystyrene granules in the aqueous isopropanol solutions results in adsorption of solvent molecules at the interparticle contacts. In addition to the adsorption of solvent molecules, consideration should be given to the second effect due to solvation forces resulted from the ﬂuid molecules. For a polystyrene granule immersed in aqueous isopropanol solution, one may consider an interparticle contact as sketched in Fig. 19(b). As discussed above, it is assumed that there was preferential adsorption of the alkyl chains of the isopropanol molecules onto the hydrophobic polystyrene particle surfaces. Hence, a hydrophilic layer was created as the hydroxyl groups of the isopropanol molecules were exposed to the solution phase. As a result, it was possible for water molecules to bind to these hydroxyl sites through hydrogen bonding to form an ordered layer of liquid structure. It is this layer of molecules from which the repulsive nature of water solvation forces arises [165]. The formation of new hydrogen bonds between the water molecules and the hydroxyl groups implies a disruption of the strong hydrogen bonding network of water to enable reorientation of the water molecules in region ‘‘A’’ (see Fig. 19(b)). Such a process is energetically unfavourable and might lead to antiparallel alignment of the water molecules in region ‘‘B’’, further away from the hydroxyl groups generating a repulsive force as shown in Fig. 19(b) [165]. Furthermore, the curved surfaces of the polystyrene particles might cause a non-uniform distribution of the number density of water molecules. The repulsive forces might be enhanced in certain regions where the number density was high. Following this hypothesis, strong hydration of the granules was manifested by the sudden rupture of the granules in 15% v/v isopropanol as the polystyrene particles were pushed apart due to the repulsive forces. In more dilute isopropanol solutions, these repulsive hydration forces could attenuate the adhesive forces between the polystyrene particles and thus reducing the granule strength. Conversely, there would not be a disruption of the hydrogen bond network when a granule was immersed in pure water since no hydrogen bonds could be established between water molecules and hydrophobic polystyrene surfaces. Hence, the attenuation in the interparticle adhesion in pure water is more appropriately interpreted as a consequence of water adsorption on the polystyrene particles through dispersion forces. This section demonstrates that the dissociation of granules is strongly dependent not only on the interaction of the particles but also the effect of the surrounding medium. For targeted performances, it may not be always possible to modify the medium and hence, the interfacial properties of the particle surfaces should be tailored to achieve the desired response of granules. Some sophisticated technologies such as reversibly wetting surfaces [168] and stimulus responsive materials [169] offer exciting potential where dispersion of granules can be triggered

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by a wide range of stimuli including pH, temperature and electric ﬁeld. The effectiveness of particle surface modiﬁcation can be accessed qualitatively using simple experiments, such as contact angle measurement.

5. CONCLUDING REMARKS The macroscopic strength of granules is a function of interparticle bonding mechanisms and the packing of the constituent particles. Three main bonding mechanisms namely autoadhesion, liquid bridges and solid bonding are reviewed with a particular emphasis on theoretical descriptions based on the properties of the constituent particles and the binder. With the advent of microscale characterisation techniques such as AFM, lateral force microscopy and nano-indentation, these theoretical treatments may be examined more quantitatively. Surprisingly, these continuum theoretical formulations were found to provide a good approximation for interactions between nanoscale contacts (see [42]). The spatial distribution of primary particles is accessible using X-ray microtomography down to the resolution of approximately several micrometers per pixel with a typical desktop tomographer [170]. However, this technique is restricted to particle and binder systems having clear contrast in the X-ray absorption. Analytical micromechanical models have highlighted the importance of fracture mechanics concepts in correlating the constituent particle properties and particle packing to the macroscopic granule strength. Furthermore, a realistic prediction of granule strength requires the non-uniform stress transmission within granules to be properly accounted for given the complexity of the rupture process. The alternative approach is the use of discrete computer simulations to examine the deformation and fracture characteristics of granules. It has been established that an external force is transmitted along many load-bearing particle chains leaving a large number of redundant interparticle contacts within a granule, which has provided a basis for some theoretical micromechanical modelling. According to DEM simulations, granules are likely to fracture through interparticle sliding across shear weakened planes created along the discrete force transmission paths within the granule dissipating energy as frictional work. It is possible to correlate granule strength parameters such as the dynamic yield stress or the coefﬁcient of restitution to the granule growth behaviour. Practically, the measurement of granule deformation under dynamic conditions provides a means of accessing the outcome of granulation for a combination of formulation and granulator. It should be pointed out that it is difﬁcult to perform experimental parametric studies on the deformation characteristic of wet granules by varying one parameter without affecting other properties of the particle-binder

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system. Perhaps, discrete particle simulation similar to that carried out by Lian et al. [74] should be pursued further by incorporating real structures of granules into DEM computations. Moreover, the appropriate control of the interfacial interaction between a liquid media and the constituent particles of granules enables granule breakage to be tailored. Hence, sophisticated surface treatments that modify such interactions deserve further investigation given the increasing demand in controlled dispersion applications. Nomenclature

A AR a a0 a1 aC c CS D3,2 E E E0 F Fb G G h0 H kn KC Mf, Md, Mr n P P1 PC Pgorge PJKR Psb Pvn

cross-sectional area of a powder assembly (m2) real area of contact (m) contact radius (m) Johnson, Kendall and Roberts (JKR) contact radius at a zero externally applied load (m) contact radius at the end of contact peeling [51] (m) critical contact radius when stable contact peeling terminates (m) critical defect size in a particle assembly (m) concentration of solid dissolved in a liquid bridge (g cm3) Sauter mean particle diameter (m) Young’s modulus (Pa) effective Young’s modulus (Pa) ensemble modulus (Pa) friction force (N) binding force between two contacting particles [99] (N) shear modulus (Pa) effective shear modulus (Pa) minimum separation distance between two spheres (m) hardness (Pa) normal contact stiffness (N m1) critical stress intensity factor (Pa m1/2) masses of feed granules, debris and residue, respectively (kg) width of fragment size distribution (–) normal applied load (N) normal force when interparticle sliding commences [51] (N) pull-off force (N) capillary force due to gorge method (N) effective JKR normal load (N) normal rupture force of a solid bridge (bond) (N) normal viscous force (N)

Mechanistic Description of Granule Deformation and Breakage

Pvt q Q r rN rsb R R s S Sc T TC u0 ue uf vr vn vrel vt V Vb Vf Vi W WA WA0 Wb Ws0 WT, WE, WM, WS

x xc Y z z0 b

1115

tangential viscous force (N) power law index of equation (38) cumulative fragment size distribution (–) local radius of curvature of a pendular liquid bridge proﬁle (m) liquid bridge neck radius (m) neck radius of a solid bridge (m) radius of a sphere or particle (m) effective radius of two spheres or particles (m) spacing factor (–) half-separation distance between a pair of spheres (m) critical half-separation for liquid bridge rupture (m) tangential applied force (N) critical tangential force when stable contact peeling terminates (N) impact velocity (ms1) elastic energy recovered during interparticle contact separation (J) fracture energy associated with interparticle contact separation (J) radial velocity of a ﬂuid under squeeze ﬂow (ms1) normal relative velocity between two spheres (ms1) relative velocity between two spheres (ms1) tangential relative velocity between two spheres (ms1) liquid bridge volume (m3) initial liquid bridge volume (m3) normal component of the critical impact failure velocity (ms1) normal impact velocity (ms1) work done or energy (J) thermodynamic work of adhesion (J m2) work of adhesion in a liquid medium (J m2) Weber number (–) total surface adhesion energy due to Derjaguin, Muller and Toporov (DMT) theory (J) total energy of a pair of spheres or particles in contact, elastic energy stored in the system, mechanical potential energy of the system and surface adhesion energy of the system, respectively (J) fragment size (m) mode size of fragments (m) dynamic yield strength (Pa) distance between two particle surfaces (A˚) equilibrium separation (A˚) half-ﬁlling angle (1)

1116

d eg em g g T, g D , g P g12 gSV, gLV, gSL gS Z j m m0 mS n pe yE rp sf ssb sy tc ti trz x+ , x xm z

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relative displacement between the centres of two contacting spheres (m) granule porosity (–) maximum compressive strain (–) surface free energy (J m2) total, dispersive and non-dispersive surface energies, respectively (J m2) interface energy between two phases (J m2) interface energies between the solid–vapour, liquid– vapour and solid–liquid interfaces, respectively (J m2) surface energy of solid in vacuum (J m2) viscosity (Pa s) contact angle (1) coefﬁcient of friction (–) ﬂow consistency coefﬁcient of static friction (–) Poisson’s ratio (–) spreading pressure (J m2) equilibrium contact angle (1) particle density (kg m3) fracture stress (Pa) fracture strength of a solid bridge (Pa) uniaxial yield stress of a granule (Pa) cohesive shear strength of powder due to Coulomb’s law (Pa) interfacial shear stress (Pa) shear stress of a power law ﬂuid (Pa) upper and lower limits of breakage extent, respectively (%) mean curvature of a pendular liquid bridge proﬁle (m1) Tabor dimensionless group (–)

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Chapter 26 A Mechanistic Description of Granule Deformation and Breakage

Chapter 26 A Mechanistic Description of Granule Deformation and Breakage

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