Transcription

1 This article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author s benefit and for the benefit of the author s institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues that you know, and providing a copy to your institution s administrator. All other uses, reproduction and distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your personal or institution s website or repository, are prohibited. For exceptions, permission may be sought for such use through Elsevier s permissions site at:

2 Chemical Engineering Science ) Abstract Adhesion of ultrafine particles A micromechanical approach Jürgen Tomas Mechanical Process Engineering, The Otto-von-Guericke-University, Universitätsplatz 2, D Magdeburg, Germany Received 9 August 2006; received in revised form 19 December 2006; accepted 21 December 2006 Available online 13 January 2007 In particle processing and product handling of fine d < 100 μm), ultrafine d < 10 μm) and nanosized particles d < 0.1 μm), the well-known flow problems of dry cohesive powders in process apparatuses or storage and transportation containers include bridging, channelling, widely spread residence time distribution associated with time consolidation or caking effects, chemical conversions and deterioration of bioparticles. Avalanching effects and oscillating mass flow rates in conveyors lead to feeding and dosing problems. Finally, insufficient apparatus and system reliability of powder processing plants are also related to these flow problems. Thus, it is very essential to understand the fundamentals of particle adhesion with respect to product quality assessment and process performance in particle technology. The state-of-the-art in constitutive modelling of elastic, elastic adhesion, elastic dissipative, plastic adhesion and plastic dissipative contact deformation response of a single isotropic contact of two smooth spheres is briefly discussed. Then the new models are shown that describe the elastic plastic force displacement and moment angle behaviour of adhesive and frictional contacts. Using the model stiff particles with soft contacts, a sphere sphere interaction of van der Waals forces without any contact deformation describes the stiff attractive term. A plate plate model is used to calculate the soft micro-contact flattening and adhesion. Various contact deformation paths for loading, unloading and contact detachment are discussed. Thus, the varying adhesion forces between particles depend directly on this frozen irreversible deformation. Thus, the adhesion force is found to be load dependent. Their essential contribution on the tangential force in an elastic plastic frictional contact with partially sticking within the contact plane and microslip, the rolling resistance and the torque of mobilized frictional contact rotation is shown Elsevier Ltd. All rights reserved. Keywords: Ultrafine particles; Particle processing; Adhesion forces; Granular materials; Contact mechanics 1. Introduction In particle processing and product handling of fine d < 100 μm), ultrafine d < 10 μm) and nanosized particles d < 0.1 μm), the well-known flow problems of dry cohesive powders in process apparatuses or storage and transportation containers include bridging, channelling, widely spread residence time distribution associated with time consolidation or caking effects, chemical conversions and deterioration of bioparticles. Avalanching effects and oscillating mass flow rates in conveyors lead to feeding and dosing problems. Finally, insufficient apparatus and system reliability of powder processing plants are also related to these adhesion problems. Tel.: ; fax: address: juergen.tomas@vst.uni-magdeburg.de. The challenge is to understand physically the fundamentals of particle adhesion with respect to product quality assessment, process performance and control in powder technology, i.e., particle conversion, formulation and handling. Therefore modelling the single particle adhesion and suitable micro macro interaction rules, constitutive equations have to be derived to describe the mechanical behaviour of cohesive powders and to simulate the dynamics of packed particle beds in powder processing e.g. agglomeration, disintegration, comminution), powder storage and flow. Using this micromechanical philosophy in powder mechanics, first physical models were derived and published by Molerus 1975, 1978) and later continued by Tomas in the theses 1983, 1991), supplemented and reviewed again 2004a,b). Independently from this, the so-called discrete element method DEM) was developed in rock mechanics Cundall and Strack, 1979). The rapid and collisional flow of /$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi: /j.ces

3 1998 J. Tomas / Chemical Engineering Science ) nonadhering particles was modelled, e.g. by Satake and Jenkins 1988) and Nedderman 1992). Tardos 1997) and Tardos et al. 2003) discussed the slow frictional and intermediate flow for compressible powders without any cohesion from the fluid mechanics point of view. Recently, a lot of papers about the flow of granular materials were compiled by Hinrichsen and Wolf 2004) and Garcia-Rojo et al. 2005). 2. State of arts Particle adhesion is caused by surface and field forces van der Waals, electrostatic and magnetic forces), material bridges between particle surfaces liquid and solid bridges, flocculants) and interlocking e.g. Israelachvili, 1992; Rumpf, 1958, 1974, 1991; Schubert, 1979, 2003; Tomas and Schubert, 1981; Tomas, 1983, 1991, 1997), Fig. 1: Surface and field forces at direct contact: Van der Waals forces all dry powders consisting of polar, induced polar and non-polar molecules, e.g. minerals, chemicals, plastics, pharmaceuticals, food). Electrostatic forces: Electric conductor metal powders). Electric non-conductor polymer powders, plastics, detergents). Magnetic force iron powder). Material bridges between particle surfaces: Hydrogen bonds of adsorbed surface layers of condensed water powders). Organic macromolecules as flocculants in suspensions in waste water). Liquid bridges of low viscous wetting liquids by capillary pressure and surface tension moist sand), high viscous bond agents resins). Solid bridges by re-crystallization of liquid bridges which contain solvents salt), solidification of swelled ultrafine gel particles starch, clay), freezing of liquid bridge bonds frozen soil), chemical reactions with adsorbed surface layers cement hydration by water) or cement with interstitial pore water concrete), solidification of high viscous bond agents asphalt), contact fusion by sintering aggregates of nanoparticles, ceramics), chemical bonds by solid solid reactions glass batch, mechanically activated metal alloys). Interlocking by macromolecular and particle shape effects: interlocking of chain branches at macromolecules proteins), interlocking of contacts by overlaps of surface asperities rough particles), interlocking by hook-like bonds fibres). It is worth to note here that van der Waals forces the focus of this paper act at the surfaces of dry ultrafine particles. They are dominant and approximately times the gravitational force, see e.g. calculations of Rumpf 1974) and Schubert 1979). The fundamentals of molecular attraction potentials and the mechanics of adhesion are treated for example by Krupp 1967), Israelachvili and Tabor 1973), Tabor 1977), Israelachvili 1992), Maugis 1999) and Kendall 2001). Contact mechanics and impact mechanics without load dependent adhesion are described by Johnson 1985) and Stronge 2000). Recently, multiscale modelling in nanomechanics, i.e., the combination of molecular dynamics MD), thermodynamics and finite element methods FEM) is described by Liu et al. 2006). Six degrees of freedom have to be considered in single contact of smooth spheres: one normal load or central impact in direction of principal axis and one torsional moment around this axis, two tangential forces and two rolling moments within the plane of flattened contact and their respective constitutive force displacement and moment angle relations, Fig. 2. The basic models for elastic behaviour were derived by Hertz 1882), Fig. 3a), and for constant adhesion by Johnson et al. 1971), Derjaguin et al. 1975), Thornton and Yin 1991), Fig. 3a). Plastic behaviour was described by Stieß 1976), Walton and Braun 1986), Thornton 1997), Thornton and Ning 1998). But the increase of adhesion force due to plastic contact deformation was introduced by Molerus 1975) and Schubert et al. 1976), Fig. 3b). Non-linear plastic, displacementdriven contact hardening was investigated by Johnson 1985), Greenwood 1997), Vu-Quoc and Zhang 1999), Mesarovic and Johnson 2000), Quintanilla 2002) and Castellanos 2005), Fig. 3c). Additionally, contact softening could be included Thomas, 2004a,b), Fig. 3c). Viscoelasticity and relaxation was considered by Yang 1966), Fig. 3d). Energy dissipation of the non-linear elastic contact with viscous spring dashpot behaviour was modelled by Kuwabara and Kono 1987), Tsuji et al. 1992) and Brilliantov et al. 1996), Fig. 3e), and during one unload/reload cycle by Sadd et al. 1993), Fig. 3f). Different elastic, elastic plastic and fully plastic behaviour without any adhesion were recently described by Stronge 2000), Fig. 3h). Time dependent viscoplasticity was modelled by Rumpf et al. 1976) and Luding et al. 2005), Fig. 3g). Using ideas of Krupp 1967), Rumpf et al. 1976), Schubert et al. 1976), Thornton and Ning 1998), the author Tomas, 2000, 2001a, 2003) had published a model for adhesive contact of ultrafine particles with elastic plastic and dissipative behaviour, Fig. 3i) and j), that is briefly explained in Section 3.1. This inelastic contact flattening leads to an increase of adhesion force F H F N ) depending on the applied normal load F N the load or pre-consolidation history. This essential effect of a soft particle contact was modelled by Krupp 1967), Dahneke 1972), Rumpf et al. 1976), Molerus 1975), Maugis

4 J. Tomas / Chemical Engineering Science ) Fig. 1. Particle adhesion and microprocesses of particle bond effects in contact. Fig. 2. Particle contact forces, moments and 6 degrees of freedom for translation and rotation. Fig. 3. Survey of constitutive models of contact deformation of smooth spherical particles in normal direction without adhesion only compression, + sign) and with adhesion tension, sign).

5 2000 J. Tomas / Chemical Engineering Science ) Constitutive models for elastic plastic, dissipative behaviour and load dependent adhesion Fig. 4. Survey of constitutive models of contact displacement of smooth spherical particles in tangential direction sign means reverse shear and displacement directions). and Pollock 1984), Tomas 2001a) and Castellanos 2005) that is shown in Section 3.2. Besides the linear elastic tangential force displacement relation, Hook s law, Fig. 4 panel a), Fromm 1927), Cattaneo 1938), Föppl 1947), Mindlin 1949), Sonntag 1950), Mindlin and Deresiewicz 1953) modelled the non-linear contact loading path up to Coulomb friction as the limit, Fig. 4 panel b). The contribution of constant adhesion forces in Coulomb friction was considered by Derjaguin et al. 1975) and Thornton 1991), Fig. 4 panel b). Mindlin and Deresiewicz 1953), Walton and Braun 1986), Thornton 1991), Di Renzo and Di Maio 2003) modelled different non-linear paths for load, unload, reload, reverse shear load, unload and load, Fig. 4 panel c), but without any adhesion. Thus, this essential effect of history or load dependent adhesion F H F N ) in Coulomb friction of ultrafine particles is demonstrated in Fig. 4 panel d) and explained in Section 3.3. The sources of an additional rolling resistance F R should be considered by partially sticking and by micro-roughness of contact surfaces, and especially, by contact deformations Fromm, 1927; Sonntag, 1950; Johnson, 1985; Iwashita and Oda, 2000). Another effect is that the sphere can rotate twist or spin) around its principal axis within the contact plane. The torque M to as radial distribution versus radius coordinate of circular elastic contact area as function of the rotation angle without any adhesion was calculated by Mindlin 1949), Cattaneo 1952), Deresiewicz 1954) and Johnson 1985). Their theories were the basis to model the influence of load dependent adhesion with respect to rolling and torsion moments of ultrafine particles, see Sections 3.4 and 3.5. It is worth to note here that normal displacement, sliding, rolling and twisting are coupled in a single contact of the moving particle packing, e.g. Farkas et al. 2003) and Brendel et al. 2004). The essential particle properties are described in Section 3.6. Finally, the sensitivity of load dependent adhesion F H F N ) on the friction limits of sliding, rolling and torsion is compared in Section 3.7. This paper is intended to focus on the model of isotropic, stiff, and linear elastic, spherical particles that are approaching to soft contacts by attractive adhesion forces of smooth surfaces. Thus, this soft or compliant contact displacement is assumed to be small h K /d>1 compared to the size diameter) of the stiff particle. The particles may have a certain material stiffness so that the volume deformation is negligible. During surface stressing the stiff particle is not so much deformed that it undergoes a certain change of the particle shape. This is equivalent to a model of healing contacts after stressing and deformation Particle contact constitutive model for normal loading A typical normal force displacement diagram is demonstrated in Fig. 5. The zero-point of this diagram h K = 0 is equivalent to the characteristic adhesion separation of a direct contact a F =0. After approaching F N a 2 with a = a F =0 h K ) from an infinite distance to a minimum separation a = a F =0, Fig. 5a), the smooth sphere sphere contact without any contact deformation is formed by the short-range attractive adhesion force F N = F H 0 the so-called jump in), F N = F H 0 = C H,slsr 1,2 6a F =0 h K ) 2 = F H 0aF 2 =0 a F =0 h K ) 2 1) with the median particle radius r 1,2 or characteristic radius of contact surface curvature r 1 and r 2 are the principal radii of surface curvatures of both spheres before flattening), 1 r 1,2 = + 1 ) 1. 2) r 1 r 2 The Hamaker constant Hamaker, 1937) C H,sls = ) J for solid liquid solid interaction index sls) according to the continuum theory of Lifshitz 1955, 1956) is related to continuous media and depends on their permittivities dielectric constants) and refractive indices, for details see Krupp 1967), Israelachvili and Tabor 1973) and Israelachvili 1992). The characteristic adhesion separation for the direct contact of spheres is of a molecular scale atomic centre-to-centre distance) and can be estimated for the total force equilibrium F = du/da = 0 of molecular attraction and repulsion potentials a = a F =0. This separation of the interaction potential minimum amounts to about a F =0 = nm. With respect to large specific surfaces of ultrafine particles, this separation a F =0 depends on the properties of liquid-equivalent packed adsorbed water layers. Consequently, the particle contact behaviour is influenced by more or less mobile adsorption layers due to condensed humidity of ambient air e.g. Chigazawa et al., 1981; Tomas, 1983; Schütz and Schubert, 1980; Restagno et al., 2002; Schumann, 2005). The minimum centre separation a F =0 is assumed to be constant during particle contact loading and unloading. This mechanical behaviour is equivalent to stiff sphere repulsion of molecules Israelachvili, 1992).

6 J. Tomas / Chemical Engineering Science ) Fig. 5. Calculated normal force displacement diagram of characteristic contact flattening of limestone particles, modelled as smooth mono-disperse spheres, median diameter d 50 = 1.2 μm. For convenience, pressure and compression are defined as positive but tension and extension are negative. The smooth spheres approach, Eq. 1), form an elastic contact, Eq. 3), start with yielding at point Y and form an elastic plastic contact, line Y U, Eq. 7), can be unloaded at point U, Eq. 12), achieve the adhesion limit at point A, Eq. 11), and finally, detach with increasing separation, Eq. 15). To avoid an overestimation, the characteristic adhesion force of rigid sphere sphere contact F H 0 = 2.64 nn was back-calculated from limestone powder shear tests Tomas, 2004a). The necessary material data for this diagram are shown in Section 3.7. Now, the contact may be loaded from point F H 0 to Y and, as the response, is elastically deformed with an approximated circular contact area Hertz, 1882), Fig. 5b), F N = 2 3 E r 1,2 h 3 K 3) with the averaged material stiffness as series of elastic elements 1 and 2 which is equivalent to the sum of reciprocal element stiffness compliances), E modulus of elasticity, ν Poisson ratio): 1 ν E 2 = ) 1 ν2 2. 4) E 1 E 2 When this Hertz curve intersects the abscissa the total force equilibrium F N = 0 is obtained within the self-equilibrating contact. With increasing external normal load this soft contact starts at a pressure p f with plastic yielding at the point Y, Fig. 5c), κ A elastic plastic contact area coefficient, κ p plastic repulsion coefficient): [ ] 3πpf κ A κ p ) 2 h K,f,el.pl = r 1,2 2E. 5) This yield point Y is located here below the abscissa, i.e., the contact force equilibrium F N = 0 includes a certain elastic or elastic plastic deformation as response of effective adhesion force 1 + κ)f H 0 with κ the elastic plastic contact consolidation coefficient. The micro-yield surface is reached and this maximum pressure p f cannot be exceeded and results in a combined elastic plastic yield limit of the plate plate contact with an annular elastic zone and a circular centre. A confined plastic field is formed inside of the contact circle. Constant mechanical properties provided, the finer the particles the smaller is the force necessary for yielding Tomas, 2000). An initial and exclusive elastic contact deformation at loading, see the position of yield point Y in Fig. 5 below the axis, has no relevance for irreversible compression of fine to ultrafine powders in the industrial practice of processing and handling. Generally, averaged macroscopic normal stresses in apparatuses, containers, conveyors and reactors amount to about σ F N /d 2?1 kpa. Because of microscopic normal forces of about F N?1 nn between the particles, exclusive elastic contact deformations can be excluded here. Besides a complex numerical model of Castellanos 2005), the linear elastic plastic force displacement models for loading introduced by Schubert et al. 1976) and by Thornton and Ning 1998) were the basis to supplement their approaches by an extended attractive force contribution of contact flattening Tomas, 2000, 2001a). The particle contact force equilibrium between attraction ) and elastic plus, simultaneously, plastic repulsion +) is calculated by rk represents the coordinate of annular elastic contact area, i.e., for r K,f rk r K): FN,i = 0 = F H 0 p VdW πrk 2 F N + p f πrk,f 2 rk + 2π p el rk )r K dr K. r K,f 6) Superposition provided Derjaguin et al., 1975), the adhesion force F H 0, Eq. 1), includes both the attraction at particle approach, i.e., short-range sphere sphere interaction, and the

7 2002 J. Tomas / Chemical Engineering Science ) contribution outside in the annular area that is located at the perimeter closed by the contact. The term with the van der Waals pressure p VdW πrk 2 models the effective attraction within the circular contact area between flattened smooth particle surfaces. The distribution of surface asperities, their local flattening and penetration is neglected at this model. Thus, the elastic plastic yield limit for loading results in a simple function, line Y U in Fig. 5: F N = πr 1,2 p f κ A κ p )h K F H 0. 7) The elastic plastic contact area coefficient κ A represents the ratio of plastic particle contact deformation area A pl to total contact deformation area A K =A pl +A el and includes a certain elastic displacement Tomas, 2001a, 2003): A pl κ A = ) 3 A K The solely elastic contact deformation A pl =0, κ A = 2 3, has only minor relevance for cohesive powders in loading, but for the complete plastic contact deformation A pl =A K ) the coefficient κ A = 1 is obtained. The adhesion force per unit planar surface area or attractive pressure p VdW which is used here to describe the van der Waals interactions at contact, is equivalent to a theoretical bond strength and can be simply calculated as σ sls mj/m 2 surface tension solid liquid solid and consequently p VdW MPa): p VdW = C H,sls 6πaF 3 = 4σ sls. 9) a =0 F =0 The plastic repulsion coefficient κ p describes a dimensionless ratio of attractive van der Waals pressure p VdW for a plate plate model, Eq. 9), to a constant repulsive micro-yield strength p f, i.e., particle micro-hardness): κ p = p VdW = C H,sls p f 6πaF 3 =0 p = 4σ sls. 10) f a F =0 p f For a stiff contact this coefficient is infinitely small, i.e., κ p 0 and for a soft or compliant contact κ p 1. For the sake of simplicity the contact parameters κ A and κ p are set constant. This makes sense in the nanometre scale of contact radius for ultrafine particles with sizes below 10 μm. The elastic plastic yield limit, Eq. 7), cannot be crossed. Between the elastic plastic yield limit and the adhesion limit F N h K,A ) = πr 1,2 p VdW h K,A F H 0 11) the elastic domain is located. Any load F N yields an increasing displacement h K. But, if one would unload, beginning at arbitrary point U, the elastically deformed, annular contact zone recovers along a parabolic curve U A, F N = 2 3 r E 1,2 h K h K,A ) 3 πr 1,2 κ p p f h K,A F H 0. 12) At the intersection A between unload curve, Eq. 12), and adhesion limit, Eq. 11), the plate plate contact remains just ad- Fig. 6. Calculated normal force displacement diagram of characteristic contact deformation of cohesive limestone particles modelled as smooth mono-disperse spheres, median diameter d 50 = 1.2 μm. The hysteresis behaviour could be shifted along the elastic plastic limit and depends on the pre-loading, or in other words, on pre-consolidation level. Thus, the variation in adhesion forces between particles depends directly on this frozen irreversible deformation, the so-called contact pre-consolidation history. hered with a frozen radius r K,A or plastic displacement h K,A and zero unload stiffness index 0) denotes the old value): h K,A,1) = h K,U 3 h K,f,el.pl h K,U + κh K,A,0) ) 2. 13) The reloading would run along an equivalent curve from point A to point U forward to the displacement h K,U as well, Fig. 5: F N = 2 3 r E 1,2 h K,U h K ) 3 + πr 1,2 p f κ A κ p )h K,U F H 0. 14) If the adhesion limit at point A in the diagram of Fig. 5 is reached then the contact plates detach with the increasing distance a = a F =0 + h K,A h K Tomas, 2003), see Fig. 5d). This actual particle separation a can be considered for the calculation by means of a long-range hyperbolic adhesion force curve F N,Z p VdW a) a 3 e.g. Krupp, 1967; Israelachvili, 1992): F N h K ) = F H 0aF 2 =0 a F =0 h K ) 2 πr 1,2 p VdW h K,A a F =0 + h K,A h K ) 3 a3 F =0. 15) It is worth to note here that the secant unload stiffness between h K,A and h K,U N/m) increases with increasing contact flattening h K,U, see the steeper unload curves in Fig. 6 with increasing load F N,U. Thus, with increasing external load applied at any process, e.g. in powder compaction, the particle contacts become stiffer and stiffer and approach the solid behaviour of a compressed briquette or tablet. These theoretical predictions of normal force displacement behaviour are very useful to describe the compression of primary elastic, elastic plastic particle compounds or

8 J. Tomas / Chemical Engineering Science ) elastic plastic granules at conveying and handling operations. The elastic plastic load and Hertzian unload curves and the unload/reload hysteresis were experimentally confirmed for coarse granules Antonyuk et al., 2005; Antonyuk, 2006) Load dependent adhesion force The slopes of elastic plastic yield and adhesion limits in Fig. 5 are characteristics of irreversible particle contact stiffness, softness or compliance. Moreover, if one eliminates the centre approach h K of the loading and unloading functions, Eqs. 7) and 12), an implied non-linear function for the contact pull-off force F N = F H at the detachment point A is obtained, see Tomas 2004a,b): F H,1) = F H 0 + κf N + F H 0 ) πr1,2 2 κ pp f [ 3F N + F H 0 ) 2r1, F ) ] 2/3 H,0) F H 0. 16) E F N + F H 0 The index 0) denotes the old and 1) the new value of iterations. The unloading point U is stored in the memory of the contact as pre-consolidation history. This general non-linear adhesion force model implies the dimensionless, elastic plastic contact consolidation coefficient κ, the stiff contribution of adhesion force F H 0 r 1,2 ) without any contact flattening and, additionally, the influence of adhesion, stiffness, averaged particle radius r 1,2 and the averaged modulus of elasticity E in the last term of the equation. This positive non-linear adhesion normal force function F H = ff N ) can be linearized Tomas, 2004a): F H = κ A κ A κ p F H 0 + κ p κ A κ p F N = 1 + κ)f H 0 + κf N. 17) Additionally, this Eq. 17) can be checked by combining the adhesion limit F H = F N h K,A ), Eq. 15), with the elastic plastic yield limit F N h K,U ), Eq. 7), for the centre approach h K,A = h K,U. The dimensionless, so-called elastic plastic contact consolidation coefficient κ determines the slope of adhesion force F H influenced by predominant plastic contact failure: κ = κ p = p VdW 1 κ A κ p p f A pl p VdW 3 A K p f. 18) This displacement or flattening coefficient κ characterizes the irreversible particle contact stiffness or softness as well. A shallow slope implies low adhesion level F H F H 0 because of stiff particle contacts, but a large slope means soft contacts, or in other words, a cohesive powder flow behaviour in the continuum or macroscale Tomas, 2001b). This model considers, additionally, the flattening of soft particle contacts caused by the adhesion force κf H 0. Thus, the total adhesion force F H consists of a stiff contribution F H 0 and a soft, displacement influenced term κf H 0 + F N ). Thus, Eq. 17) can be interpreted as a general linear constitutive model, i.e., linear in forces, but non-linear in material characteristics Tomas, 2001a, 2004) Elastic plastic, frictional tangential force of load dependent adhesive contact The tangential stiffness within the contact plane for elastic frictional behaviour can be derived according to the theory of Mindlin 1949) as function of tangential displacement δ without any adhesion force. Mindlin s model is supplemented for the elastic plastic contact flattening caused by a sufficiently large normal force F N and, consequently, partially sticking by load dependent adhesion force F H F N ), Eq. 17), the index H is used here for the adhesive contact of ultrafine particles 1 and 2): k T,H = df T dδ = 4G r K 1 δ ) 1/2. 19) δ C,H G i = E i /21 + ν i ), i = 1, 2 is the shear modulus and the averaged shear modulus G is written as sum of reciprocal element stiffness see Eq. 4) for E as well): 2 G ν1 = ν ) ) G 1 G 2 The initial stiffness k T,H0 is found at tangential displacement δ = 0: k T,H0 = df T dδ = 4G r K = 4G r 1,2 h K. 21) δ=0 Thus, the tangential force displacement relation within the contact plane can be expressed by a linear elastic contribution for the no-slip region within the contact area according to Hook s law, Fig. 4 panel a): F T = 4G r K δ. 22) It is worth to note here that the contact radius r K depends on the applied normal force F N of the elastic plastic yield limit, Eq. 7): rk 2 = F N + F H 0 πp f κ A κ p ). 23) The tangential force displacement relation F T δ) can be expressed by a linear elastic contribution for the no-slip region within the contact plane according to Hook s law. F T and the initial tangential stiffness k T,H0 F N ), Eq. 21), increase with increasing contact flattening F H F N ), see Fig. 7: F T = 4G r K δ = 4G 1 + κ)f H 0 + F N ) δ. 24) πκ A p f The Coulomb friction limit of the tangential force is described by the coefficient of internal friction μ i. It depends on the elastic plastic contact consolidation, i.e., elastic plastic flattening by the normal force F N and the variable adhesion force F H F N ) as well, Eq. 17): F T,C,H = μ i [F N + F H F N )]=μ i 1 + κ)f H 0 + F N ). 25) The contact loses its elastic tangential stiffness at k T,H = 0 and completely mobilized contact sliding is obtained for the friction

9 2004 J. Tomas / Chemical Engineering Science ) Fig. 7. Tangential force within contact plane) displacement diagram for limestone particles modelled as smooth mono-disperse spheres median d 50,3 =1.2 μm, shear modulus G =34 kn/mm 2, friction coefficient μ i =0.76). Compared to the contact radius r K = 6 14 nm, the elastic range is very small and limited by the tangential displacement δ C,H < 0.06 nm. At these Coulomb friction limits grey points, index C) the elastic behaviour is transmitted into the frictional behaviour of contact sliding shown by constant tangential force F T,C,H = ff N,F H F N )) = fδ). Adhesion force parameters are the elastic plastic contact consolidation coefficient κ = and the positive) adhesion force without any contact flattening F H 0 =2.64 nn. The non-linear elastic hysteresis curves by partially sticking and microslip within the contact plane, Eqs. 28) 30), are not shown here. limit of displacement δ = δ C,H under load dependent adhesion force F H F N ), Eq. 17), see grey points for yield in Fig. 7: δ C,H = 3μ i1 + κ)f H 0 + F N ) 8G r K = 3μ i πκa 8G p f 1 + κ)f H 0 + F N ). 26) For checking the consistency of this friction limit, combining Eqs. 22), 23) and 25), the transition between the linear elastic range and contact failure by Coulomb friction is calculated by the friction limit of tangential displacement δ 0,H : δ 0,H = μ i[f N + F H F N )] 4G r K F N ) = μ i πκa 4G p f 1 + κ)f H 0 + F N ). 27) Both friction limits, Eqs. 26) and 27) are geometrically similar δ C,H = 1.5δ 0,H and increase with increasing load F N. Mindlin s 1949) non-linear model F T = fδ) is supplemented here for the first tangential loading of a pre-consolidated or flattened elastic plastic contact: F T = F T,C,H [1 1 δ ) ] 3/2. 28) δ C,H Thus, the extended functions for unloading and shear in reverse direction sign) are F T = F T,U 2F T,C,H [1 1 δ ) ] U δ 3/2 29) 2δ C,H and for reloading in the previous +) shear direction for F T,reload = F T,U and δ reload = δ U : F T = F T,reload + 2F T,C,H [1 1 δ + δ ) ] 3/2 reload. 2δ C,H 30) Using theses constitutive force displacement models for elastic plastic, frictional behaviour, Eqs. 29) and 30), the dissipated mechanical work per one closed cycle between unload δ U and reload δ reload displacements can be calculated. This energy consumption will be discussed in a next paper Elastic plastic, frictional rolling resistance of load dependent adhesive contact Johnson s creep model 1985) for elastic frictional behaviour without adhesion is the basis for the extension by elastic plastic contact flattening and consequently, partially sticking by load dependent adhesion force F H F N ). The stiffness k R,H of rolling resistance F R at the centre of mass can be expressed as function of relative rolling angle γ of both spheres index H for adhesion of ultrafine particles): k R,H = df R dγ = 16G 1 + κ)f H 0 + F N ) 1 γ ) 2, 31) π4 3ν) κ A p f γ C,H ) ) δ1 δ2 γ = tan tan δ 1 δ 2. 32) r 1 r 2 r 1 r 2 The initial stiffness k R,H0 is found at zero of rolling angle γ=0: k R,H0 = df R dγ = 16G 1 + κ)f H 0 + F N ). 33) γ=0 π4 3ν) κ A p f The contact loses its elastic rolling stiffness at k R,H = 0 and the rolling friction limit with respect to a critical rolling angle γ = γ C,H is obtained, γ C,H = 3α R4 3ν) 16Gr 1,2 πκa p f 1 + κ)f N + F H 0 ). 34) This friction limit γ C,H is geometrically similar to a tangent limit γ 0,H calculated by ratio of force limit F R,C,H and initial stiffness k R,H0, i.e., γ C,H =3γ 0,H =3F R,C,H /k R,H0. It increases with increasing load F N like the friction limit of tangential displacement δ C,H, Eq. 26). The frictional rolling resistance force of smooth soft spheres acts at the centre of mass and can be considered by a tilting moment relation of the force pair around the grey pivot at the perimeter of contact circle. As the response of contact flattening a lever arm of contact radius r K with respect to F N is generated which is equilibrated by rolling resistance F R acting perpendicular to direction of F N with the lever arm r h K /2, see in Fig. 8 the rectangular lines of panel right. With Eq. 17) for elastic plastic contact displacement in normal direction and

10 J. Tomas / Chemical Engineering Science ) Fig. 8. Rolling resistance force at the centre of mass) rolling angle diagram for limestone particles modelled as smooth mono-disperse spheres. The so-called rolling friction coefficient is load dependent μ R F N ) = r K F N )/r and amounts here to μ R The linear elastic range is very small restricted by the limit of rolling angle γ C,H = At these limits the elastic behaviour is transmitted into the contact rolling shown by constant rolling resistance force F R,C,H = fγ), Eq. 36). F R,C,H and γ C,H depend on adhesion force F H F N ). The hysteresis by partially sticking and microslip within the contact plane, Eqs. 38) 40), is demonstrated here only for the largest normal force. load dependent adhesion force F H F N ), the critical rolling resistance F R,C,H results in F R,C,H = r K r [F N + F H F N )]= r K r 1 + κ)f H 0 + F N ). 35) Substitution of the contact radius r K of the elastic plastic yield limit, Eq. 23), results nearly in a proportionality as F R,C,H F N ) F 3/2 N : F R,C,H = μ R 1 + κ)f H 0 + F N ) = 1 + κ)3 F H 0 + F N ) 3 πr1,2 2 p. 36) f κ A With the moment torque) M R = F R r of a sphere with radius r the friction limit of the rolling moment follows from Eq. 36): 1 + κ) 3 F H 0 + F N ) 3 M R,C,H =. 37) πp f κ A By integration of Eq. 31) the force rolling angle function for the first loading of the flattened elastic plastic contact is obtained, F R = F R,C,H [1 1 γ ) ] 3. 38) γ C,H Similarly to the constitutive tangential force displacement model, Eqs. 29) and 30), the function for unloading and rolling in reverse direction sign) is obtained as F R = F R,U 2F R,C,H [1 1 γ ) ] U γ 3 39) 2γ C,H and for reloading in the previous +) rolling direction, i.e., F R,reload = F R,U and γ reload = γ U : F R = F R,reload + 2F R,C,H [1 1 γ ) ] reload + γ 3. 2γ C,H 40) These force displacement models F R γ) models are shown in Fig. 8. Equivalent to the resistance forces the rolling moment angle relations M R γ) = F R γ)r for loading, unloading and reloading result in M R = M R,C,H [1 1 γ ) ] 3, 41) γ C,H M R = M R,U 2M R,C,H [1 1 γ ) ] U γ 3, 42) 2γ C,H M R = M R,reload + 2M R,C,H [1 1 γ ) ] 3 reload+γ. 43) 2γ C,H These rolling moment rolling angle relations M R γ) were derived to calculate the dissipated mechanical work per one closed cycle between unload γ U and reload γ reload angles Elastic plastic, frictional twisting of load dependent adhesive contact The torsional stiffness for elastic frictional behaviour without any adhesion force can be derived according to Deresiewicz 1954) as function of torsional moment M to. But the torsional stiffness k to,h for elastic plastic contact flattening and consequently, partially sticking by load dependent adhesion force F H F N ), Eq. 17), amounts to index H for adhesion of ultrafine particles): [ k to,h = dm to d = 8Gr3 K 2 1 M to 3 M to,c,h ) 1/2 1 ] 1. 44) The initial stiffness k to,h 0 is found at vanishing torsional moment M to = 0 with Eq. 23) for the contact radius r K : k to,h 0 = dm to d = 8G [ ] 1 + κ)fh 0 + F N ) 3/2. 45) Mto =0 3 πκ A p f This initial stiffness k to,h 0, Eq. 45), increases with the load dependent adhesion force F H F N ) and differs from the initial torsional stiffness according to Mindlin 1949). For the limestone powder d 50 = 1.2 μm) it gives very small values k to,h 0 = ) Nm. For simplicity, the elastic moment rotation angle function can be approached as linear relation, Fig. 9. The contact loses its elastic torsional stiffness at k to,h = 0 and completely mobilized frictional contact rotation is obtained for the friction limit of moment M to = M to,c,h under

11 2006 J. Tomas / Chemical Engineering Science ) under load dependent adhesion force F H F N ), Eq. 17): C,H = 3μ i1 + κ)f H 0 + F N ) 4Gr 2. 50) K Fig. 9. Torsional moment rotation angle diagram for limestone particles modelled as smooth mono-disperse spheres. The linear elastic range is very small and restricted by a constant friction limit of rotation angle C,H = At this Coulomb friction limit C,H the elastic behaviour is transmitted into the contact sliding shown by constant torque M to,c,h = f ), Eq. 46). Only M to,c,h depends on adhesion force F H F N ). The non-linear elastic hysteresis curves by partially sticking and microslip, Eqs. 49), 53) and 54), are not shown here. load dependent adhesion force F H F N ) with Eq. 17): M to,c,h = 2μ i1 + κ)f H 0 + F N )r K 3 = 2μ i 1 + κ) 3 F N + F H 0 ) 3. 46) 3 πκ A p f If one integrates the reciprocal torsional stiffness k to,h compliance), Eq. 44), Mto = k to,h Mto ) 1 dmto 0 = 3 [ Mto 8Gr M to K 0 M to,c,h ) 1/2 1] dmto 47) the relative rotation angle = 1 2 is obtained as function of torsional moment M to with the corresponding friction limits M to,c,h, Eq. 46), and C,H, Eq. 51): = 4 3 C,H [ 1 4 M ) ] to 1 M to. 48) 4 M to,c,h M to,c,h Rearranging Eq. 48) and the torsional moment M to ) results in a function of rotation angle for the first torsional loading of the flattened elastic plastic contact: [ ] 3 M to = 4M to,c,h ) C,H 4 C,H Unfortunately, Deresiewicz 1954) equations 17), 23) and 25) could not be applied here for ultrafine particles. With the moment of completely mobilized frictional contact rotation M to,c,h, Eq. 46), results in the critical rotation angle C,H Substituting the contact radius r K by Eq. 23) yields the simple constitutive relation for the frictional limit of rotation of an elastic plastic contact with adhesion: C,H = 3πμ iκ A p f. 51) 4G This friction limit C,H is geometrically similar to the tangent limit 0,H calculated by ratio of moment limit M to,c,h and initial stiffness k to,h 0 : C,H = 3 0,H = 3 M to,c,h. 52) k to,h 0 It is independent of the load F N provided that κ A = const.) and depends only on the particle properties. This critical value amounts to C,H for limestone particles, Fig. 9. This new model for loading of an elastic plastic frictional contact, Eq. 49) may be supplemented by the models for unloading, i.e., twisting in the reverse direction, [ 3 M to = M to,u 8M U ) to,c,h C,H ] 3 U ) 1 53) 8 C,H and for reloading, i.e., twisting in the previous direction: [ 3 M to = M to,reload + 8M reload + ) to,c,h C,H ] 3 reload + ) 1. 54) 8 C,H Using theses constitutive torsional moment rotation angle functions for elastic plastic, frictional behaviour, Eqs. 53) and 54), the dissipated mechanical work per one closed cycle between unload U and reload reload angles can be calculated. This load dependent energy consumption will be discussed in a next paper as well Essential constitutive particle parameters From these elastic plastic and frictional force displacement laws F N h K ), F T δ), force force equations F H F N ) and moment angle models M R γ), M to ) one can conclude that six independent mechanical, i.e., physically based, material parameters are necessary to know for the micromechanics of particle adhesion: 1) The characteristic particle size d 50 or, more precise, the median radius of contact surface curvature before flattening r 1,2, Eq. 2), is necessary to know.

12 J. Tomas / Chemical Engineering Science ) ) The Hamaker constant C H,sls is the characteristic material parameter of the van der Waals adhesion between practically dry particles for both types of bonds the sphere sphere interaction of contact points F H 0 and the plate plate interaction of flattened contact by an external normal load F N macroscopic normal stress σ) and the internal adhesion effect F H F N ). From this value the attractive van der Waals pressure p VdW and the surface tension σ sls, Eq. 9), can be obtained. 3) The plastic micro-yield strength p f characterizes the plastic repulsion behaviour of the sphere contact. The ratio of the attractive van der Waals force per unit planar surface p VdW and this repulsive plastic micro-yield strength p f results in the dimensionless plastic contact repulsion coefficient κ p, Eq. 10), and from that the elastic plastic contact consolidation coefficient κ is obtained. 4) The elastic contact repulsion is determined by the modulus of elasticity E and 5) by the Poisson ratio ν. From these material data the shear modulus G is calculated. 6) The frictional contact failure, i.e., the loss of elastic tangential or torsional stiffness, is characterized by the coefficient of internal friction μ i between the adhesive particle surfaces. This information is essential to model the plastic yield limits of cohesive powders when the stresses are approaching the maximum shear stress the Coulomb Mohr yield conditions for incipient and steady-state flow and consolidation Comparison of load dependent friction limits With these effective material data for ideally assumed, smooth mono-disperse limestone particles, the following diagram Fig. 10 is calculated to compare the sensitivity of load dependent adhesion F H F N ) on the friction limits of sliding, rolling and torsion. Following material data are used: 1) Median particle diameter d 50,3 = 1.2 μm. 2) Hamaker constant C H,sls = J, equilibrium centre separation for dipole interaction a F =0 = nm. The characteristic adhesion force of rigid sphere sphere contact F H 0 = 2.64 nn was back-calculated from powder shear tests. 3) The plastic micro-yield strength p f = 300 N/mm 2, elastic plastic contact area coefficient κ A = 5 6, plastic repulsion coefficient κ p = 0.153, elastic plastic contact consolidation coefficient κ = ) Modulus of elasticity E = 150 kn/mm 2. 5) Poisson ratio ν = 0.28, shear modulus G = 34 kn/mm 2 and 6) Contact friction coefficient μ i = The tangential force or maximum resistance F T,C,H to let the contact begin to slide is larger than to pull-off and separate the particles by maximum adhesion force F H. The microscopic Fig. 10. Force and moment normal force diagram of limestone particles to compare the load dependent elastic plastic adhesion limit F H as positive force and the load dependent elastic plastic friction limits of tangential force F T,C,H, rolling moment M R,C,H and torsional moment M to,c,h, Eqs. 17), 25), 37) and 46). rolling moment or maximum resistance M R,C,H to let roll the contact is larger than to let rotate or twist the particles by maximum torsional moment M to,c,h in the macroscopic low pressure range of 1 50 kpa applied in powder handling. This microscopic load dependent adhesion effect on the friction limits of single particle contacts leads macroscopically to the significant influence of pre-consolidation stress on the slow frictional flow of ultrafine cohesive powders that is well known in powder handling practice, see e.g. Tomas 2004a,b). 4. Summary and conclusions By the model stiff particles with soft contacts and the contact force equilibrium, universal equations were derived which include the elastic plastic particle contact behaviour with adhesion, load unload hysteresis and a history dependent adhesion force function. With increasing adhesion force the normal, tangential, rolling and torsional contact stiffness and the Coulomb friction limits increase. Six independent mechanical material parameters are necessary to know for this micromechanical system of constitutive equations for particle adhesion: characteristic particle radius r 1,2 of surface curvature, Hamaker constant C H,sls as the characteristic adhesion parameter, micro-yield strength p f for plastic repulsion, modulus of elasticity E and Poisson ratio ν as the elastic repulsion parameters and coefficient of internal friction μ i. The consequences of elastic dissipative as well as elastic plastic, frictional unloading and reloading paths of normal and tangential forces and rolling and torsional moments for the load dependent energy absorption and the friction work will be discussed and compared in a next paper. This microscopic energy consumption is meaningful for the macroscopic frictional shear

13 2008 J. Tomas / Chemical Engineering Science ) flow of cohesive ultrafine particulate solids and the agglomerate disintegration in powder processing and handling. The microscopic load dependent adhesion effect on the friction limits of single particle contacts leads macroscopically to the significant influence of pre-consolidation stress on the frictional flow of ultrafine cohesive powders that is well known in powder handling practice. Thus, the physical models derived here are used for the advanced data evaluation of various powder product properties concerning particle size distribution nanoparticles to granules), moisture content dry, moist and wet) and material properties minerals, chemicals, pigments, waste, plastics, food, etc.). For example, this approach was applied to evaluate the data during compression and impact tests of granules Antonyuk et al., 2005, 2006). The models shown here are very meaningful to describe the frictional shear flow of cohesive ultrafine powders. For this purpose, all the normal and tangential forces as well as angular and trajectorial moments of particles have to be balanced to simulate their dynamics with physically realistic material data by the DEM Tykhoniuk et al., 2007). By modelling the single particle adhesion and by suitable micro macro interaction rules, constitutive equations will be derived to describe the pre-consolidation dependent, mechanical behaviour of ultrafine cohesive powders and to simulate the dynamics of packed particle beds in powder processing, i.e., at agglomeration, disintegration, size reduction, powder storage and flow. At present, with this advanced knowledge an improved apparatus design process chambers, feed hoppers, silos and conveyors) is accomplished for industrial partners in process industries. Notation a contact separation, nm a F =0 minimum centre separation for force equilibrium of molecular attraction and repulsion potentials, nm A S,m mass related surface area, m 2 /g C H,sls Hamaker constant according to Lifshitz theory for solid liquid solid interaction with liquid-like adsorption layers, J d particle size, μm d 50,3 median particle diameter, mass basis, μm E modulus of elasticity, kn/mm 2 F force, N F H adhesion force of single contact, nn F H 0 adhesion force of a rigid contact without any contact deformation, nn F N normal force of single contact, nn F T tangential force of single contact, nn G shear modulus, kn/mm 2 h K height of overlap, indentation or centre approach of two spheres, nm k N contact stiffness in normal direction, N/m k R contact stiffness of rolling resistance, μn torsional contact stiffness, Nm k to k T M M R M to p f p VdW r r K contact stiffness in tangential direction, N/m moment torque), Nm rolling moment, Nm torsional moment torque), Nm plastic micro-yield strength of particle contact, MPa attractive van der Waals pressure, see Eq. 9), MPa particle radius, μm contact radius, nm Greek letters γ rolling angle, distortion, dimensionless δ tangential contact displacement, nm κ elastic plastic contact consolidation coefficient, dimensionless κ A elastic plastic contact area coefficient, dimensionless κ p plastic repulsion coefficient, dimensionless μ i coefficient of internal particle particle friction, i.e., Coulomb friction, dimensionless ν Poisson ratio, dimensionless σ sls surface tension solid liquid solid, mj/m 2 rotation angle, dimensionless Indices A detachment C Coulomb friction el elastic H adhesion K particle contact l liquid m mass related N normal p pressure related pl plastic reload reload R rolling s solid to torsional T tangential U unload VdW van der Waals 0 initial, zero point 0) old value of iterations 1,2 particle 1, particle 2 3 mass basis of cumulative distribution of particle size d 3 ) 50 median particle size, i.e., 50% of cumulative distribution Acknowledgements The author would like to thank H. Altenbach, A. Bertam, U. Gabbert, K. Kassner, D. Regener, P. Streitenberger, L. Tobiska, G. Warnecke, M. Zehn from Mechanics, Physics,

14 J. Tomas / Chemical Engineering Science ) Material Sciences, Mathematics and Process Engineering for the fruitful collaboration within the DFG graduate school 828 Micro Macro Interactions in Structured Media and Particle Systems. The advices from H.-J. Butt, M. Kappl and S. Luding with respect to the fundamentals of particle and powder mechanics were especially appreciated during the collaboration of the project shear dynamics of cohesive, fine-disperse particle systems of the joint research program Behaviour of Granular Media of German Research Association DFG). References Antonyuk, S., Deformations- und Bruchverhalten von kugelförmigen Granulaten bei Druck- und Stoßbeanspruchung. Ph.D. Thesis, Docupoint, Magdeburg. Antonyuk, S., Tomas, J., Heinrich, S., Mörl, L., Breakage behaviour of spherical granulates by compression. Chemical Engineering Science 60, Antonyuk, S., Tomas, J., Heinrich, S., Mörl, L., Impact breakage of spherical granules: experimental study and DEM simulation. Chemical Engineering and Processing 45, Brendel, L., Unger, T., Wolf, D.E., Contact dynamics for beginners. In: Hinrichsen, H., Wolf, D.E. Eds.), The Physics of Granular Media. Wiley-VCH, Berlin, pp Brilliantov, N.V., Spahn, V., Hertzsch, J.M., Pöschel, T., Model for collisions in granular gases. Physical Review E 53, Castellanos, A., The relationship between attractive interparticle forces and bulk behavior in dry and uncharged fine powders. Advances in Physics 54, Cattaneo, C., Meccanica Sul contatto di due corpi elastici, distribuzione locale degli sforsi. Academia Nationale dei Lincei, Rendiconti 27, , , Cattaneo, C., Sulla torsione die due sfere elastiche a contatto. Annali della Scuola Normale Superiore di Pisa Seria III) 6, 1 16 In: Deresiewicz, H. Ed.), Contact of elastic spheres under an oscillating torsional couple. Transactions of American Society Mechanical Engineers, Journal of Applied Mechanics 21, Chigazawa, M., Yamaguchi, T., Kanazava, T., Adhesion mechanism of powder particles due to vapour adsorption. In: Proceedings of International Symposium on Powder Technology, Kyoto, pp Cundall, P.A., Strack, O.D.L., A discrete numerical model for granular assemblies. Geotechnique 29, Dahneke, B., The influence of flattening on the adhesion of particles. Journal of Colloid and Interface Science 40, Deresiewicz, H., Contact of elastic spheres under an oscillating torsional couple. Transactions of American Society Mechanical Engineers, Journal of Applied Mechanics 21, Derjaguin, B.V., Muller, V.M., Toporov, Y.P., Effect of contact deformations on the adhesion of particles. Journal of Colloid and Interface Science 53, Di Renzo, A., Di Maio, F.P., Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes. Chemical Engineering Science 59, Farkas, Z., Bartels, G., Unger, T., Wolf, D.E., Frictional coupling between sliding and spinning motion. Physical Review Letter 90, Föppl, L., Die strenge Lösung für die rollende Reibung. Leibnitz Verlag, München. Fromm, G., Berechnung des Schlupfes beim Rollen deformierbarer Scheiben. Zeitschrift für Angewandte Mathematik und Mechanik 7, Garcia-Rojo, R., Herrmann, H.J., McNamara, S. Eds.), Powder and Grains 2005, vols. 1 and 2. A.A. Balkema Publishers, Leiden. Greenwood, J.A., Adhesion of elastic spheres. Proceedings of Royal Society A 453, Hamaker, H.C., The London van der Waals attraction between spherical particles. Physica 4, Hertz, H., Über die Berührung fester elastischer Körper. Journal für die Reine und Angewandte Mathematik 92, Hinrichsen, H., Wolf, D.E. Eds.), 2004.The Physics of Granular Media. Wiley-VCH, Berlin. Israelachvili, J.N., Intermolecular and Surface Forces. Academic Press, London. Israelachvili, J.N., Tabor, D., Van der Waals forces: theory and experiment. Progress of Surface and Membrane Science 7, Iwashita, K., Oda, M., Micro-deformation mechanism of shear banding process based on modified distinct element method. Powder Technology 109, Johnson, K.L., Contact Mechanics. Cambridge University Press, Cambridge. Johnson, K.L., Kendall, K., Roberts, A.D., Surface energy and the contact of elastic solids. Proceedings of Royal Society A 324, Kendall, K., Molecular Adhesion and its Application. Kluwer Academic Publishers/Plenum Publishers, New York. Krupp, H., Particle adhesion theory and experiment. Advances in Colloid and Interface Science 1, Kuwabara, C., Kono, K., Restitution coefficient in a collision between two spheres. Japanese Journal of Applied Physics 26, Lifshitz, E.M., Teorija moleculyarnych sil prityasheniya meshdu tverdymi telami. Journal of Experimental and Theoretical Physics USSR 29, Lifshitz, E.M., The theory of molecular attractive forces between solids. Soviet Physics Journal of Experimental and Theoretical Physics 2, Liu, W.K., Karpov, E.G., Park, H.S., Nano Mechanics and Materials: Theory, Multiscale Methods and Applications. Wiley, Chichester. Luding, S., Manetsberger, K., Müller, J., A discrete model for long time sintering. Journal of Mechanics and Physics of Solids 53, Maugis, D., Contact, Adhesion and Rupture of Elastic Solids. Springer, Berlin. Maugis, D., Pollock, H.M., Surface forces, deformation and adherence at metal microcontacts. Acta Metallica 32, Mesarovic, S.D., Johnson, K.L., Adhesive contact of elastic plastic spheres. Journal of Mechanics and Physics of Solids 48, Mindlin, R.D., Compliance of elastic bodies in contact. Transactions of American Society Mechanical Engineers, Journal of Applied Mechanics 16, Mindlin, R.D., Deresiewicz, H., Elastic spheres in contact under varying oblique forces. Transactions of American Society Mechanical Engineers, Journal of Applied Mechanics 20, Molerus, O., Theory of yield of cohesive powders. Powder Technology 12, Molerus, O., Effect of interparticle cohesive forces on the flow behaviour of powders. Powder Technology 20, Nedderman, R.M., Statics and Kinematics of Granular Materials. Cambridge University Press, Cambridge. Quintanilla, M.A.S., Mechanica de medios granulares cohesivos y su relacion con las fuerzas entre particulas. Ph.D. Thesis, Universidad de Sevilla. Restagno, F., Crassous, J., Cottin-Bizonne, C., Charlaix, E., Adhesion between weakly rough beads. Physical Review E 65, Rumpf, H., Grundlagen und Methoden des Granulierens. Chemie- Ingenieur-Technik 30, Rumpf, H., Die Wissenschaft des Agglomerierens. Chemie-Ingenieur- Technik 46, Rumpf, H., Particle Technology. Chapman & Hall, London. Rumpf, H., Sommer, K., Steier, K., Mechanismen des Haftkraftverstärkung bei der Partikelhaftung durch plastische Verformen, Sintern und viskoelastisches Fließen. Chemie-Ingenieur-Technik 48, Sadd, M., Tai, Q., Shukla, A., Contact law effects on wave propagation in particulate materials using distinct element modelling. International Journal of Non-Linear Mechanics 28,